Runtime Complexity of the query pattern
rev(g,a)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^2)).

(0) Obligation:

Clauses:

rev([], []).
rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys)).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Query: rev(g,a)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

rev([], []).
app([], X, X).
rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys)).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Query: rev(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];
1 [label="1: rev(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: rev(T1, T2), SCOPE: 1, CLAUSE: 0 | rev(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | rev([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: rev(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nrev(T1, T2) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: rev([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(rev(T10, X14), app(X14, .(T9, []), T12))\n\nT9 is ground\nT10 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: rev(T10, X14)\n\nT10 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: app(T13, .(T9, []), T12)\n\nT9 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: rev(T10, X14), SCOPE: 2, CLAUSE: 0 | rev(T10, X14), SCOPE: 2, CLAUSE: 2\n\nT10 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true | rev([], X14), SCOPE: 2, CLAUSE: 2\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: rev(T10, X14), SCOPE: 2, CLAUSE: 2\n\nT10 is ground\nX14 is free\nrev(T10, X14) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: rev([], X14), SCOPE: 2, CLAUSE: 2\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(rev(T19, X30), app(X30, .(T18, []), X31))\n\nT18 is ground\nT19 is ground\nX31 is free\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: rev(T19, X30)\n\nT19 is ground\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: app(T20, .(T18, []), X31)\n\nT18 is ground\nT20 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: app(T20, .(T18, []), X31), SCOPE: 3, CLAUSE: 1 | app(T20, .(T18, []), X31), SCOPE: 3, CLAUSE: 3\n\nT18 is ground\nT20 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true | app([], .(T25, []), X31), SCOPE: 3, CLAUSE: 3\n\nT25 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: app(T20, .(T18, []), X31), SCOPE: 3, CLAUSE: 3\n\nT18 is ground\nT20 is ground\nX31 is free\nX36 is free\napp(T20, .(T18, []), X31) !=? app([], X36, X36)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: app([], .(T25, []), X31), SCOPE: 3, CLAUSE: 3\n\nT25 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: app(T33, .(T34, []), X55)\n\nT33 is ground\nT34 is ground\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: app(T13, .(T9, []), T12), SCOPE: 4, CLAUSE: 1 | app(T13, .(T9, []), T12), SCOPE: 4, CLAUSE: 3\n\nT9 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: true | app([], .(T41, []), T12), SCOPE: 4, CLAUSE: 3\n\nT41 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: app(T13, .(T9, []), T12), SCOPE: 4, CLAUSE: 3\n\nT9 is ground\nT13 is ground\nX62 is free\napp(T13, .(T9, []), T12) !=? app([], X62, X62)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: app([], .(T41, []), T12), SCOPE: 4, CLAUSE: 3\n\nT41 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: app(T51, .(T52, []), T54)\n\nT51 is ground\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 74 [arrowhead = none ];
74 -> 4;

75 [label="EVAL with clause\nrev([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 75 [arrowhead = none ];
75 -> 5;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 76 [arrowhead = none ];
76 -> 7;

77 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 77 [arrowhead = none ];
77 -> 9;

78 [label="EVAL with clause\nrev(.(X11, X12), X13) :- ','(rev(X12, X14), app(X14, .(X11, []), X13)).\nand substitutionX11 -> T9,\nX12 -> T10,\nT1 -> .(T9, T10),\nT2 -> T12,\nX13 -> T12,\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 78 [arrowhead = none ];
78 -> 15;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 79 [arrowhead = none ];
79 -> 17;

80 [label="BACKTRACK\nfor clause: rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 80 [arrowhead = none ];
80 -> 11;

81 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 81 [arrowhead = none ];
81 -> 22;

82 [label="SPLIT 2\nnew knowledge:\nT10 is ground\nT13 is ground\nreplacements:X14 -> T13", fontsize=14, style = filled, fillcolor = violet];
15 -> 82 [arrowhead = none ];
82 -> 23;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 83 [arrowhead = none ];
83 -> 28;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 84 [arrowhead = none ];
84 -> 62;

85 [label="EVAL with clause\nrev([], []).\nand substitutionT10 -> [],\nX14 -> []", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 85 [arrowhead = none ];
85 -> 30;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 86 [arrowhead = none ];
86 -> 32;

87 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 87 [arrowhead = none ];
87 -> 33;

88 [label="EVAL with clause\nrev(.(X27, X28), X29) :- ','(rev(X28, X30), app(X30, .(X27, []), X29)).\nand substitutionX27 -> T18,\nX28 -> T19,\nT10 -> .(T18, T19),\nX14 -> X31,\nX29 -> X31", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 88 [arrowhead = none ];
88 -> 36;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 89 [arrowhead = none ];
89 -> 37;

90 [label="BACKTRACK\nfor clause: rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 90 [arrowhead = none ];
90 -> 35;

91 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
36 -> 91 [arrowhead = none ];
91 -> 40;

92 [label="SPLIT 2\nnew knowledge:\nT19 is ground\nT20 is ground\nreplacements:X30 -> T20", fontsize=14, style = filled, fillcolor = violet];
36 -> 92 [arrowhead = none ];
92 -> 41;

93 [label="INSTANCE with matching:\nT10 -> T19\nX14 -> X30", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 93 [arrowhead = none , style=dashed];
93 -> 22[style=dashed];

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 94 [arrowhead = none ];
94 -> 48;

95 [label="EVAL with clause\napp([], X36, X36).\nand substitutionT20 -> [],\nT18 -> T25,\nX36 -> .(T25, []),\nX31 -> .(T25, [])", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 95 [arrowhead = none ];
95 -> 50;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 96 [arrowhead = none ];
96 -> 51;

97 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 97 [arrowhead = none ];
97 -> 52;

98 [label="EVAL with clause\napp(.(X51, X52), X53, .(X51, X54)) :- app(X52, X53, X54).\nand substitutionX51 -> T32,\nX52 -> T33,\nT20 -> .(T32, T33),\nT18 -> T34,\nX53 -> .(T34, []),\nX54 -> X55,\nX31 -> .(T32, X55)", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 98 [arrowhead = none ];
98 -> 58;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 99 [arrowhead = none ];
99 -> 59;

100 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 100 [arrowhead = none ];
100 -> 53;

101 [label="INSTANCE with matching:\nT20 -> T33\nT18 -> T34\nX31 -> X55", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 101 [arrowhead = none , style=dashed];
101 -> 41[style=dashed];

102 [label="EVAL with clause\napp([], X62, X62).\nand substitutionT13 -> [],\nT9 -> T41,\nX62 -> .(T41, []),\nT12 -> .(T41, [])", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 102 [arrowhead = none ];
102 -> 66;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 103 [arrowhead = none ];
103 -> 67;

104 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 104 [arrowhead = none ];
104 -> 68;

105 [label="EVAL with clause\napp(.(X75, X76), X77, .(X75, X78)) :- app(X76, X77, X78).\nand substitutionX75 -> T50,\nX76 -> T51,\nT13 -> .(T50, T51),\nT9 -> T52,\nX77 -> .(T52, []),\nX78 -> T54,\nT12 -> .(T50, T54),\nT53 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 105 [arrowhead = none ];
105 -> 72;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 106 [arrowhead = none ];
106 -> 73;

107 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 107 [arrowhead = none ];
107 -> 69;

108 [label="INSTANCE with matching:\nT13 -> T51\nT9 -> T52\nT12 -> T54", fontsize=14, style = filled, fillcolor = lightgrey];
72 -> 108 [arrowhead = none , style=dashed];
108 -> 23[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1([])
f1_in(.(z0, z1)) → U1(f15_in(z1, z0), .(z0, z1))
U1(f15_out1(z0, z1), .(z2, z3)) → f1_out1(z1)
f23_in([], z0) → f23_out1(.(z0, []))
f23_in(.(z0, z1), z2) → U2(f23_in(z1, z2), .(z0, z1), z2)
U2(f23_out1(z0), .(z1, z2), z3) → f23_out1(.(z1, z0))
f22_in([]) → f22_out1([])
f22_in(.(z0, z1)) → U3(f36_in(z1, z0), .(z0, z1))
U3(f36_out1(z0, z1), .(z2, z3)) → f22_out1(z1)
f41_in([], z0) → f41_out1(.(z0, []))
f41_in(.(z0, z1), z2) → U4(f41_in(z1, z2), .(z0, z1), z2)
U4(f41_out1(z0), .(z1, z2), z3) → f41_out1(.(z1, z0))
f15_in(z0, z1) → U5(f22_in(z0), z0, z1)
U5(f22_out1(z0), z1, z2) → U6(f23_in(z0, z2), z1, z2, z0)
U6(f23_out1(z0), z1, z2, z3) → f15_out1(z3, z0)
f36_in(z0, z1) → U7(f22_in(z0), z0, z1)
U7(f22_out1(z0), z1, z2) → U8(f41_in(z0, z2), z1, z2, z0)
U8(f41_out1(z0), z1, z2, z3) → f36_out1(z3, z0)

Tuples:

F1_IN(.(z0, z1)) → c1(U1'(f15_in(z1, z0), .(z0, z1)), F15_IN(z1, z0))
F23_IN(.(z0, z1), z2) → c4(U2'(f23_in(z1, z2), .(z0, z1), z2), F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(U3'(f36_in(z1, z0), .(z0, z1)), F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(U4'(f41_in(z1, z2), .(z0, z1), z2), F41_IN(z1, z2))
F15_IN(z0, z1) → c12(U5'(f22_in(z0), z0, z1), F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c13(U6'(f23_in(z0, z2), z1, z2, z0), F23_IN(z0, z2))
F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
U7'(f22_out1(z0), z1, z2) → c16(U8'(f41_in(z0, z2), z1, z2, z0), F41_IN(z0, z2))

S tuples:

F1_IN(.(z0, z1)) → c1(U1'(f15_in(z1, z0), .(z0, z1)), F15_IN(z1, z0))
F23_IN(.(z0, z1), z2) → c4(U2'(f23_in(z1, z2), .(z0, z1), z2), F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(U3'(f36_in(z1, z0), .(z0, z1)), F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(U4'(f41_in(z1, z2), .(z0, z1), z2), F41_IN(z1, z2))
F15_IN(z0, z1) → c12(U5'(f22_in(z0), z0, z1), F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c13(U6'(f23_in(z0, z2), z1, z2, z0), F23_IN(z0, z2))
F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
U7'(f22_out1(z0), z1, z2) → c16(U8'(f41_in(z0, z2), z1, z2, z0), F41_IN(z0, z2))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f23_in, U2, f22_in, U3, f41_in, U4, f15_in, U5, U6, f36_in, U7, U8

Defined Pair Symbols:

F1_IN, F23_IN, F22_IN, F41_IN, F15_IN, U5', F36_IN, U7'

Compound Symbols:

c1, c4, c7, c10, c12, c13, c15, c16

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

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1([])
f1_in(.(z0, z1)) → U1(f15_in(z1, z0), .(z0, z1))
U1(f15_out1(z0, z1), .(z2, z3)) → f1_out1(z1)
f23_in([], z0) → f23_out1(.(z0, []))
f23_in(.(z0, z1), z2) → U2(f23_in(z1, z2), .(z0, z1), z2)
U2(f23_out1(z0), .(z1, z2), z3) → f23_out1(.(z1, z0))
f22_in([]) → f22_out1([])
f22_in(.(z0, z1)) → U3(f36_in(z1, z0), .(z0, z1))
U3(f36_out1(z0, z1), .(z2, z3)) → f22_out1(z1)
f41_in([], z0) → f41_out1(.(z0, []))
f41_in(.(z0, z1), z2) → U4(f41_in(z1, z2), .(z0, z1), z2)
U4(f41_out1(z0), .(z1, z2), z3) → f41_out1(.(z1, z0))
f15_in(z0, z1) → U5(f22_in(z0), z0, z1)
U5(f22_out1(z0), z1, z2) → U6(f23_in(z0, z2), z1, z2, z0)
U6(f23_out1(z0), z1, z2, z3) → f15_out1(z3, z0)
f36_in(z0, z1) → U7(f22_in(z0), z0, z1)
U7(f22_out1(z0), z1, z2) → U8(f41_in(z0, z2), z1, z2, z0)
U8(f41_out1(z0), z1, z2, z3) → f36_out1(z3, z0)

Tuples:

F23_IN(.(z0, z1), z2) → c4(U2'(f23_in(z1, z2), .(z0, z1), z2), F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(U3'(f36_in(z1, z0), .(z0, z1)), F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(U4'(f41_in(z1, z2), .(z0, z1), z2), F41_IN(z1, z2))
F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F1_IN(.(z0, z1)) → c(U1'(f15_in(z1, z0), .(z0, z1)))
F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(U6'(f23_in(z0, z2), z1, z2, z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c(U8'(f41_in(z0, z2), z1, z2, z0))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))

S tuples:

F23_IN(.(z0, z1), z2) → c4(U2'(f23_in(z1, z2), .(z0, z1), z2), F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(U3'(f36_in(z1, z0), .(z0, z1)), F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(U4'(f41_in(z1, z2), .(z0, z1), z2), F41_IN(z1, z2))
F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F1_IN(.(z0, z1)) → c(U1'(f15_in(z1, z0), .(z0, z1)))
F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(U6'(f23_in(z0, z2), z1, z2, z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c(U8'(f41_in(z0, z2), z1, z2, z0))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f23_in, U2, f22_in, U3, f41_in, U4, f15_in, U5, U6, f36_in, U7, U8

Defined Pair Symbols:

F23_IN, F22_IN, F41_IN, F36_IN, F1_IN, F15_IN, U5', U7'

Compound Symbols:

c4, c7, c10, c15, c

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

Removed 6 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1([])
f1_in(.(z0, z1)) → U1(f15_in(z1, z0), .(z0, z1))
U1(f15_out1(z0, z1), .(z2, z3)) → f1_out1(z1)
f23_in([], z0) → f23_out1(.(z0, []))
f23_in(.(z0, z1), z2) → U2(f23_in(z1, z2), .(z0, z1), z2)
U2(f23_out1(z0), .(z1, z2), z3) → f23_out1(.(z1, z0))
f22_in([]) → f22_out1([])
f22_in(.(z0, z1)) → U3(f36_in(z1, z0), .(z0, z1))
U3(f36_out1(z0, z1), .(z2, z3)) → f22_out1(z1)
f41_in([], z0) → f41_out1(.(z0, []))
f41_in(.(z0, z1), z2) → U4(f41_in(z1, z2), .(z0, z1), z2)
U4(f41_out1(z0), .(z1, z2), z3) → f41_out1(.(z1, z0))
f15_in(z0, z1) → U5(f22_in(z0), z0, z1)
U5(f22_out1(z0), z1, z2) → U6(f23_in(z0, z2), z1, z2, z0)
U6(f23_out1(z0), z1, z2, z3) → f15_out1(z3, z0)
f36_in(z0, z1) → U7(f22_in(z0), z0, z1)
U7(f22_out1(z0), z1, z2) → U8(f41_in(z0, z2), z1, z2, z0)
U8(f41_out1(z0), z1, z2, z3) → f36_out1(z3, z0)

Tuples:

F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
U7'(f22_out1(z0), z1, z2) → c

S tuples:

F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
U7'(f22_out1(z0), z1, z2) → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, f23_in, U2, f22_in, U3, f41_in, U4, f15_in, U5, U6, f36_in, U7, U8

Defined Pair Symbols:

F36_IN, F1_IN, F15_IN, U5', U7', F23_IN, F22_IN, F41_IN

Compound Symbols:

c15, c, c4, c7, c10, c

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
U5'(f22_out1(z0), z1, z2) → c

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1([])
f1_in(.(z0, z1)) → U1(f15_in(z1, z0), .(z0, z1))
U1(f15_out1(z0, z1), .(z2, z3)) → f1_out1(z1)
f23_in([], z0) → f23_out1(.(z0, []))
f23_in(.(z0, z1), z2) → U2(f23_in(z1, z2), .(z0, z1), z2)
U2(f23_out1(z0), .(z1, z2), z3) → f23_out1(.(z1, z0))
f22_in([]) → f22_out1([])
f22_in(.(z0, z1)) → U3(f36_in(z1, z0), .(z0, z1))
U3(f36_out1(z0, z1), .(z2, z3)) → f22_out1(z1)
f41_in([], z0) → f41_out1(.(z0, []))
f41_in(.(z0, z1), z2) → U4(f41_in(z1, z2), .(z0, z1), z2)
U4(f41_out1(z0), .(z1, z2), z3) → f41_out1(.(z1, z0))
f15_in(z0, z1) → U5(f22_in(z0), z0, z1)
U5(f22_out1(z0), z1, z2) → U6(f23_in(z0, z2), z1, z2, z0)
U6(f23_out1(z0), z1, z2, z3) → f15_out1(z3, z0)
f36_in(z0, z1) → U7(f22_in(z0), z0, z1)
U7(f22_out1(z0), z1, z2) → U8(f41_in(z0, z2), z1, z2, z0)
U8(f41_out1(z0), z1, z2, z3) → f36_out1(z3, z0)

Tuples:

F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
U7'(f22_out1(z0), z1, z2) → c

S tuples:

F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))
U7'(f22_out1(z0), z1, z2) → c

K tuples:

F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c

Defined Rule Symbols:

f1_in, U1, f23_in, U2, f22_in, U3, f41_in, U4, f15_in, U5, U6, f36_in, U7, U8

Defined Pair Symbols:

F36_IN, F1_IN, F15_IN, U5', U7', F23_IN, F22_IN, F41_IN

Compound Symbols:

c15, c, c4, c7, c10, c

(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.

F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))

We considered the (Usable) Rules:

f22_in([]) → f22_out1([])
f22_in(.(z0, z1)) → U3(f36_in(z1, z0), .(z0, z1))
f36_in(z0, z1) → U7(f22_in(z0), z0, z1)
U3(f36_out1(z0, z1), .(z2, z3)) → f22_out1(z1)
U7(f22_out1(z0), z1, z2) → U8(f41_in(z0, z2), z1, z2, z0)
f41_in([], z0) → f41_out1(.(z0, []))
f41_in(.(z0, z1), z2) → U4(f41_in(z1, z2), .(z0, z1), z2)
U8(f41_out1(z0), z1, z2, z3) → f36_out1(z3, z0)
U4(f41_out1(z0), .(z1, z2), z3) → f41_out1(.(z1, z0))

And the Tuples:

F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
U7'(f22_out1(z0), z1, z2) → c

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

POL(.(x₁, x₂)) = [1] + x₁ + x₂   
POL(F15_IN(x₁, x₂)) = [3] + [3]x₁ + [2]x₂   
POL(F1_IN(x₁)) = [1] + [3]x₁   
POL(F22_IN(x₁)) = [1] + [3]x₁   
POL(F23_IN(x₁, x₂)) = [1]   
POL(F36_IN(x₁, x₂)) = [3] + [3]x₁ + [3]x₂   
POL(F41_IN(x₁, x₂)) = [2]x₂   
POL(U3(x₁, x₂)) = 0   
POL(U4(x₁, x₂, x₃)) = 0   
POL(U5'(x₁, x₂, x₃)) = [2] + x₃   
POL(U7(x₁, x₂, x₃)) = [3] + [2]x₁ + [2]x₂ + x₃   
POL(U7'(x₁, x₂, x₃)) = [3]x₃   
POL(U8(x₁, x₂, x₃, x₄)) = [2]x₁ + x₂ + x₃   
POL([]) = 0   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c15(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(f22_in(x₁)) = 0   
POL(f22_out1(x₁)) = [2]   
POL(f36_in(x₁, x₂)) = [3] + [2]x₁ + [2]x₂   
POL(f36_out1(x₁, x₂)) = [3] + x₂   
POL(f41_in(x₁, x₂)) = 0   
POL(f41_out1(x₁)) = [2] + x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1([])
f1_in(.(z0, z1)) → U1(f15_in(z1, z0), .(z0, z1))
U1(f15_out1(z0, z1), .(z2, z3)) → f1_out1(z1)
f23_in([], z0) → f23_out1(.(z0, []))
f23_in(.(z0, z1), z2) → U2(f23_in(z1, z2), .(z0, z1), z2)
U2(f23_out1(z0), .(z1, z2), z3) → f23_out1(.(z1, z0))
f22_in([]) → f22_out1([])
f22_in(.(z0, z1)) → U3(f36_in(z1, z0), .(z0, z1))
U3(f36_out1(z0, z1), .(z2, z3)) → f22_out1(z1)
f41_in([], z0) → f41_out1(.(z0, []))
f41_in(.(z0, z1), z2) → U4(f41_in(z1, z2), .(z0, z1), z2)
U4(f41_out1(z0), .(z1, z2), z3) → f41_out1(.(z1, z0))
f15_in(z0, z1) → U5(f22_in(z0), z0, z1)
U5(f22_out1(z0), z1, z2) → U6(f23_in(z0, z2), z1, z2, z0)
U6(f23_out1(z0), z1, z2, z3) → f15_out1(z3, z0)
f36_in(z0, z1) → U7(f22_in(z0), z0, z1)
U7(f22_out1(z0), z1, z2) → U8(f41_in(z0, z2), z1, z2, z0)
U8(f41_out1(z0), z1, z2, z3) → f36_out1(z3, z0)

Tuples:

F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
U7'(f22_out1(z0), z1, z2) → c

S tuples:

U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))
U7'(f22_out1(z0), z1, z2) → c

K tuples:

F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))

Defined Rule Symbols:

f1_in, U1, f23_in, U2, f22_in, U3, f41_in, U4, f15_in, U5, U6, f36_in, U7, U8

Defined Pair Symbols:

F36_IN, F1_IN, F15_IN, U5', U7', F23_IN, F22_IN, F41_IN

Compound Symbols:

c15, c, c4, c7, c10, c

(13) CdtKnowledgeProof (EQUIVALENT transformation)

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

U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1([])
f1_in(.(z0, z1)) → U1(f15_in(z1, z0), .(z0, z1))
U1(f15_out1(z0, z1), .(z2, z3)) → f1_out1(z1)
f23_in([], z0) → f23_out1(.(z0, []))
f23_in(.(z0, z1), z2) → U2(f23_in(z1, z2), .(z0, z1), z2)
U2(f23_out1(z0), .(z1, z2), z3) → f23_out1(.(z1, z0))
f22_in([]) → f22_out1([])
f22_in(.(z0, z1)) → U3(f36_in(z1, z0), .(z0, z1))
U3(f36_out1(z0, z1), .(z2, z3)) → f22_out1(z1)
f41_in([], z0) → f41_out1(.(z0, []))
f41_in(.(z0, z1), z2) → U4(f41_in(z1, z2), .(z0, z1), z2)
U4(f41_out1(z0), .(z1, z2), z3) → f41_out1(.(z1, z0))
f15_in(z0, z1) → U5(f22_in(z0), z0, z1)
U5(f22_out1(z0), z1, z2) → U6(f23_in(z0, z2), z1, z2, z0)
U6(f23_out1(z0), z1, z2, z3) → f15_out1(z3, z0)
f36_in(z0, z1) → U7(f22_in(z0), z0, z1)
U7(f22_out1(z0), z1, z2) → U8(f41_in(z0, z2), z1, z2, z0)
U8(f41_out1(z0), z1, z2, z3) → f36_out1(z3, z0)

Tuples:

F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
U7'(f22_out1(z0), z1, z2) → c

S tuples:

F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))

K tuples:

F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c

Defined Rule Symbols:

f1_in, U1, f23_in, U2, f22_in, U3, f41_in, U4, f15_in, U5, U6, f36_in, U7, U8

Defined Pair Symbols:

F36_IN, F1_IN, F15_IN, U5', U7', F23_IN, F22_IN, F41_IN

Compound Symbols:

c15, c, c4, c7, c10, c

(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.

F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))

We considered the (Usable) Rules:

f22_in([]) → f22_out1([])
f22_in(.(z0, z1)) → U3(f36_in(z1, z0), .(z0, z1))
f36_in(z0, z1) → U7(f22_in(z0), z0, z1)
U3(f36_out1(z0, z1), .(z2, z3)) → f22_out1(z1)
U7(f22_out1(z0), z1, z2) → U8(f41_in(z0, z2), z1, z2, z0)
f41_in([], z0) → f41_out1(.(z0, []))
f41_in(.(z0, z1), z2) → U4(f41_in(z1, z2), .(z0, z1), z2)
U8(f41_out1(z0), z1, z2, z3) → f36_out1(z3, z0)
U4(f41_out1(z0), .(z1, z2), z3) → f41_out1(.(z1, z0))

And the Tuples:

F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
U7'(f22_out1(z0), z1, z2) → c

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F15_IN(x₁, x₂)) = [3]x₁   
POL(F1_IN(x₁)) = [3]x₁   
POL(F22_IN(x₁)) = 0   
POL(F23_IN(x₁, x₂)) = x₁   
POL(F36_IN(x₁, x₂)) = 0   
POL(F41_IN(x₁, x₂)) = 0   
POL(U3(x₁, x₂)) = x₁   
POL(U4(x₁, x₂, x₃)) = [1] + x₁   
POL(U5'(x₁, x₂, x₃)) = x₁   
POL(U7(x₁, x₂, x₃)) = [1] + x₁   
POL(U7'(x₁, x₂, x₃)) = 0   
POL(U8(x₁, x₂, x₃, x₄)) = x₁   
POL([]) = 0   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c15(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(f22_in(x₁)) = [2]x₁   
POL(f22_out1(x₁)) = x₁   
POL(f36_in(x₁, x₂)) = [2] + [2]x₁   
POL(f36_out1(x₁, x₂)) = x₂   
POL(f41_in(x₁, x₂)) = [1] + x₁   
POL(f41_out1(x₁)) = x₁   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1([])
f1_in(.(z0, z1)) → U1(f15_in(z1, z0), .(z0, z1))
U1(f15_out1(z0, z1), .(z2, z3)) → f1_out1(z1)
f23_in([], z0) → f23_out1(.(z0, []))
f23_in(.(z0, z1), z2) → U2(f23_in(z1, z2), .(z0, z1), z2)
U2(f23_out1(z0), .(z1, z2), z3) → f23_out1(.(z1, z0))
f22_in([]) → f22_out1([])
f22_in(.(z0, z1)) → U3(f36_in(z1, z0), .(z0, z1))
U3(f36_out1(z0, z1), .(z2, z3)) → f22_out1(z1)
f41_in([], z0) → f41_out1(.(z0, []))
f41_in(.(z0, z1), z2) → U4(f41_in(z1, z2), .(z0, z1), z2)
U4(f41_out1(z0), .(z1, z2), z3) → f41_out1(.(z1, z0))
f15_in(z0, z1) → U5(f22_in(z0), z0, z1)
U5(f22_out1(z0), z1, z2) → U6(f23_in(z0, z2), z1, z2, z0)
U6(f23_out1(z0), z1, z2, z3) → f15_out1(z3, z0)
f36_in(z0, z1) → U7(f22_in(z0), z0, z1)
U7(f22_out1(z0), z1, z2) → U8(f41_in(z0, z2), z1, z2, z0)
U8(f41_out1(z0), z1, z2, z3) → f36_out1(z3, z0)

Tuples:

F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
U7'(f22_out1(z0), z1, z2) → c

S tuples:

F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))

K tuples:

F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))

Defined Rule Symbols:

f1_in, U1, f23_in, U2, f22_in, U3, f41_in, U4, f15_in, U5, U6, f36_in, U7, U8

Defined Pair Symbols:

F36_IN, F1_IN, F15_IN, U5', U7', F23_IN, F22_IN, F41_IN

Compound Symbols:

c15, c, c4, c7, c10, c

(17) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))

We considered the (Usable) Rules:

f22_in([]) → f22_out1([])
f22_in(.(z0, z1)) → U3(f36_in(z1, z0), .(z0, z1))
f36_in(z0, z1) → U7(f22_in(z0), z0, z1)
U3(f36_out1(z0, z1), .(z2, z3)) → f22_out1(z1)
U7(f22_out1(z0), z1, z2) → U8(f41_in(z0, z2), z1, z2, z0)
f41_in([], z0) → f41_out1(.(z0, []))
f41_in(.(z0, z1), z2) → U4(f41_in(z1, z2), .(z0, z1), z2)
U8(f41_out1(z0), z1, z2, z3) → f36_out1(z3, z0)
U4(f41_out1(z0), .(z1, z2), z3) → f41_out1(.(z1, z0))

And the Tuples:

F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
U7'(f22_out1(z0), z1, z2) → c

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F15_IN(x₁, x₂)) = [1] + x₁ + x₁²   
POL(F1_IN(x₁)) = x₁²   
POL(F22_IN(x₁)) = x₁²   
POL(F23_IN(x₁, x₂)) = 0   
POL(F36_IN(x₁, x₂)) = [1] + x₁ + x₁²   
POL(F41_IN(x₁, x₂)) = [1] + x₁   
POL(U3(x₁, x₂)) = [1] + x₁   
POL(U4(x₁, x₂, x₃)) = [1] + x₁   
POL(U5'(x₁, x₂, x₃)) = 0   
POL(U7(x₁, x₂, x₃)) = x₁   
POL(U7'(x₁, x₂, x₃)) = x₁   
POL(U8(x₁, x₂, x₃, x₄)) = x₁   
POL([]) = 0   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c15(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(f22_in(x₁)) = [1] + x₁   
POL(f22_out1(x₁)) = [1] + x₁   
POL(f36_in(x₁, x₂)) = [1] + x₁   
POL(f36_out1(x₁, x₂)) = x₂   
POL(f41_in(x₁, x₂)) = [1] + x₁   
POL(f41_out1(x₁)) = x₁   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1([])
f1_in(.(z0, z1)) → U1(f15_in(z1, z0), .(z0, z1))
U1(f15_out1(z0, z1), .(z2, z3)) → f1_out1(z1)
f23_in([], z0) → f23_out1(.(z0, []))
f23_in(.(z0, z1), z2) → U2(f23_in(z1, z2), .(z0, z1), z2)
U2(f23_out1(z0), .(z1, z2), z3) → f23_out1(.(z1, z0))
f22_in([]) → f22_out1([])
f22_in(.(z0, z1)) → U3(f36_in(z1, z0), .(z0, z1))
U3(f36_out1(z0, z1), .(z2, z3)) → f22_out1(z1)
f41_in([], z0) → f41_out1(.(z0, []))
f41_in(.(z0, z1), z2) → U4(f41_in(z1, z2), .(z0, z1), z2)
U4(f41_out1(z0), .(z1, z2), z3) → f41_out1(.(z1, z0))
f15_in(z0, z1) → U5(f22_in(z0), z0, z1)
U5(f22_out1(z0), z1, z2) → U6(f23_in(z0, z2), z1, z2, z0)
U6(f23_out1(z0), z1, z2, z3) → f15_out1(z3, z0)
f36_in(z0, z1) → U7(f22_in(z0), z0, z1)
U7(f22_out1(z0), z1, z2) → U8(f41_in(z0, z2), z1, z2, z0)
U8(f41_out1(z0), z1, z2, z3) → f36_out1(z3, z0)

Tuples:

F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
U7'(f22_out1(z0), z1, z2) → c

S tuples:none
K tuples:

F1_IN(.(z0, z1)) → c(F15_IN(z1, z0))
F15_IN(z0, z1) → c(U5'(f22_in(z0), z0, z1))
F15_IN(z0, z1) → c(F22_IN(z0))
U5'(f22_out1(z0), z1, z2) → c(F23_IN(z0, z2))
F1_IN(.(z0, z1)) → c
U5'(f22_out1(z0), z1, z2) → c
F36_IN(z0, z1) → c15(U7'(f22_in(z0), z0, z1), F22_IN(z0))
F22_IN(.(z0, z1)) → c7(F36_IN(z1, z0))
U7'(f22_out1(z0), z1, z2) → c(F41_IN(z0, z2))
U7'(f22_out1(z0), z1, z2) → c
F23_IN(.(z0, z1), z2) → c4(F23_IN(z1, z2))
F41_IN(.(z0, z1), z2) → c10(F41_IN(z1, z2))

Defined Rule Symbols:

f1_in, U1, f23_in, U2, f22_in, U3, f41_in, U4, f15_in, U5, U6, f36_in, U7, U8

Defined Pair Symbols:

F36_IN, F1_IN, F15_IN, U5', U7', F23_IN, F22_IN, F41_IN

Compound Symbols:

c15, c, c4, c7, c10, c

(19) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(20) BOUNDS(O(1), O(1))

(21) 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: rev(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: rev(T1, T2), SCOPE: 1, CLAUSE: 0 | rev(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | rev([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: rev(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nrev(T1, T2) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: rev([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(rev(T7, X10), app(X10, .(T6, []), T9))\n\nT6 is ground\nT7 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(rev(T7, X10), app(X10, .(T6, []), T9)), SCOPE: 2, CLAUSE: 0 | ','(rev(T7, X10), app(X10, .(T6, []), T9)), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nT7 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app([], .(T6, []), T9) | ','(rev([], X10), app(X10, .(T6, []), T9)), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(rev(T7, X10), app(X10, .(T6, []), T9)), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nT7 is ground\nX10 is free\nrev(T7, X10) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app([], .(T6, []), T9)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(rev([], X10), app(X10, .(T6, []), T9)), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: app([], .(T6, []), T9), SCOPE: 3, CLAUSE: 1 | app([], .(T6, []), T9), SCOPE: 3, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true | app([], .(T16, []), T9), SCOPE: 3, CLAUSE: 3\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: app([], .(T6, []), T9), SCOPE: 3, CLAUSE: 3\n\nT6 is ground\nX17 is free\napp([], .(T6, []), T9) !=? app([], X17, X17)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app([], .(T16, []), T9), SCOPE: 3, CLAUSE: 3\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(','(rev(T22, X41), app(X41, .(T21, []), X42)), app(X42, .(T6, []), T9))\n\nT6 is ground\nT21 is ground\nT22 is ground\nX42 is free\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: rev(T22, X41)\n\nT22 is ground\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(app(T23, .(T21, []), X42), app(X42, .(T6, []), T9))\n\nT6 is ground\nT21 is ground\nT23 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: rev(T22, X41), SCOPE: 4, CLAUSE: 0 | rev(T22, X41), SCOPE: 4, CLAUSE: 2\n\nT22 is ground\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: true | rev([], X41), SCOPE: 4, CLAUSE: 2\n\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: rev(T22, X41), SCOPE: 4, CLAUSE: 2\n\nT22 is ground\nX41 is free\nrev(T22, X41) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: rev([], X41), SCOPE: 4, CLAUSE: 2\n\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(rev(T29, X58), app(X58, .(T28, []), X59))\n\nT28 is ground\nT29 is ground\nX59 is free\nX58 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: rev(T29, X58)\n\nT29 is ground\nX58 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: app(T30, .(T28, []), X59)\n\nT28 is ground\nT30 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: app(T30, .(T28, []), X59), SCOPE: 5, CLAUSE: 1 | app(T30, .(T28, []), X59), SCOPE: 5, CLAUSE: 3\n\nT28 is ground\nT30 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true | app([], .(T35, []), X59), SCOPE: 5, CLAUSE: 3\n\nT35 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: app(T30, .(T28, []), X59), SCOPE: 5, CLAUSE: 3\n\nT28 is ground\nT30 is ground\nX59 is free\nX64 is free\napp(T30, .(T28, []), X59) !=? app([], X64, X64)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: app([], .(T35, []), X59), SCOPE: 5, CLAUSE: 3\n\nT35 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: app(T43, .(T44, []), X83)\n\nT43 is ground\nT44 is ground\nX83 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: app(T23, .(T21, []), X42)\n\nT21 is ground\nT23 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: app(T49, .(T6, []), T9)\n\nT6 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: app(T49, .(T6, []), T9), SCOPE: 6, CLAUSE: 1 | app(T49, .(T6, []), T9), SCOPE: 6, CLAUSE: 3\n\nT6 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: true | app([], .(T56, []), T9), SCOPE: 6, CLAUSE: 3\n\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: app(T49, .(T6, []), T9), SCOPE: 6, CLAUSE: 3\n\nT6 is ground\nT49 is ground\nX94 is free\napp(T49, .(T6, []), T9) !=? app([], X94, X94)", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: app([], .(T56, []), T9), SCOPE: 6, CLAUSE: 3\n\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: app(T66, .(T67, []), T69)\n\nT66 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 83 [arrowhead = none ];
83 -> 3;

84 [label="EVAL with clause\nrev([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 84 [arrowhead = none ];
84 -> 6;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 85 [arrowhead = none ];
85 -> 8;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 86 [arrowhead = none ];
86 -> 10;

87 [label="EVAL with clause\nrev(.(X7, X8), X9) :- ','(rev(X8, X10), app(X10, .(X7, []), X9)).\nand substitutionX7 -> T6,\nX8 -> T7,\nT1 -> .(T6, T7),\nT2 -> T9,\nX9 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 87 [arrowhead = none ];
87 -> 13;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 88 [arrowhead = none ];
88 -> 14;

89 [label="BACKTRACK\nfor clause: rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 89 [arrowhead = none ];
89 -> 12;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 90 [arrowhead = none ];
90 -> 16;

91 [label="EVAL with clause\nrev([], []).\nand substitutionT7 -> [],\nX10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 91 [arrowhead = none ];
91 -> 18;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 92 [arrowhead = none ];
92 -> 19;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 93 [arrowhead = none ];
93 -> 20;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 94 [arrowhead = none ];
94 -> 21;

95 [label="EVAL with clause\nrev(.(X38, X39), X40) :- ','(rev(X39, X41), app(X41, .(X38, []), X40)).\nand substitutionX38 -> T21,\nX39 -> T22,\nT7 -> .(T21, T22),\nX10 -> X42,\nX40 -> X42", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 95 [arrowhead = none ];
95 -> 38;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 96 [arrowhead = none ];
96 -> 39;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 97 [arrowhead = none ];
97 -> 24;

98 [label="BACKTRACK\nfor clause: rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 98 [arrowhead = none ];
98 -> 34;

99 [label="EVAL with clause\napp([], X17, X17).\nand substitutionT6 -> T16,\nX17 -> .(T16, []),\nT9 -> .(T16, [])", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 99 [arrowhead = none ];
99 -> 25;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 100 [arrowhead = none ];
100 -> 26;

101 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 101 [arrowhead = none ];
101 -> 27;

102 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 102 [arrowhead = none ];
102 -> 31;

103 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 103 [arrowhead = none ];
103 -> 29;

104 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
38 -> 104 [arrowhead = none ];
104 -> 42;

105 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nT23 is ground\nreplacements:X41 -> T23", fontsize=14, style = filled, fillcolor = violet];
38 -> 105 [arrowhead = none ];
105 -> 43;

106 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 106 [arrowhead = none ];
106 -> 44;

107 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
43 -> 107 [arrowhead = none ];
107 -> 74;

108 [label="SPLIT 2\nnew knowledge:\nT23 is ground\nT21 is ground\nT49 is ground\nreplacements:X42 -> T49", fontsize=14, style = filled, fillcolor = violet];
43 -> 108 [arrowhead = none ];
108 -> 75;

109 [label="EVAL with clause\nrev([], []).\nand substitutionT22 -> [],\nX41 -> []", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 109 [arrowhead = none ];
109 -> 45;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 110 [arrowhead = none ];
110 -> 46;

111 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 111 [arrowhead = none ];
111 -> 47;

112 [label="EVAL with clause\nrev(.(X55, X56), X57) :- ','(rev(X56, X58), app(X58, .(X55, []), X57)).\nand substitutionX55 -> T28,\nX56 -> T29,\nT22 -> .(T28, T29),\nX41 -> X59,\nX57 -> X59", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 112 [arrowhead = none ];
112 -> 54;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 113 [arrowhead = none ];
113 -> 55;

114 [label="BACKTRACK\nfor clause: rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 114 [arrowhead = none ];
114 -> 49;

115 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
54 -> 115 [arrowhead = none ];
115 -> 56;

116 [label="SPLIT 2\nnew knowledge:\nT29 is ground\nT30 is ground\nreplacements:X58 -> T30", fontsize=14, style = filled, fillcolor = violet];
54 -> 116 [arrowhead = none ];
116 -> 57;

117 [label="INSTANCE with matching:\nT22 -> T29\nX41 -> X58", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 117 [arrowhead = none , style=dashed];
117 -> 42[style=dashed];

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 118 [arrowhead = none ];
118 -> 60;

119 [label="EVAL with clause\napp([], X64, X64).\nand substitutionT30 -> [],\nT28 -> T35,\nX64 -> .(T35, []),\nX59 -> .(T35, [])", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 119 [arrowhead = none ];
119 -> 61;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 120 [arrowhead = none ];
120 -> 63;

121 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 121 [arrowhead = none ];
121 -> 64;

122 [label="EVAL with clause\napp(.(X79, X80), X81, .(X79, X82)) :- app(X80, X81, X82).\nand substitutionX79 -> T42,\nX80 -> T43,\nT30 -> .(T42, T43),\nT28 -> T44,\nX81 -> .(T44, []),\nX82 -> X83,\nX59 -> .(T42, X83)", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 122 [arrowhead = none ];
122 -> 70;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 123 [arrowhead = none ];
123 -> 71;

124 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 124 [arrowhead = none ];
124 -> 65;

125 [label="INSTANCE with matching:\nT30 -> T43\nT28 -> T44\nX59 -> X83", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 125 [arrowhead = none , style=dashed];
125 -> 57[style=dashed];

126 [label="INSTANCE with matching:\nT30 -> T23\nT28 -> T21\nX59 -> X42", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 126 [arrowhead = none , style=dashed];
126 -> 57[style=dashed];

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
75 -> 127 [arrowhead = none ];
127 -> 76;

128 [label="EVAL with clause\napp([], X94, X94).\nand substitutionT49 -> [],\nT6 -> T56,\nX94 -> .(T56, []),\nT9 -> .(T56, [])", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 128 [arrowhead = none ];
128 -> 77;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 129 [arrowhead = none ];
129 -> 78;

130 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 130 [arrowhead = none ];
130 -> 79;

131 [label="EVAL with clause\napp(.(X107, X108), X109, .(X107, X110)) :- app(X108, X109, X110).\nand substitutionX107 -> T65,\nX108 -> T66,\nT49 -> .(T65, T66),\nT6 -> T67,\nX109 -> .(T67, []),\nX110 -> T69,\nT9 -> .(T65, T69),\nT68 -> T69", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 131 [arrowhead = none ];
131 -> 81;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 132 [arrowhead = none ];
132 -> 82;

133 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 133 [arrowhead = none ];
133 -> 80;

134 [label="INSTANCE with matching:\nT49 -> T66\nT6 -> T67\nT9 -> T69", fontsize=14, style = filled, fillcolor = lightgrey];
81 -> 134 [arrowhead = none , style=dashed];
134 -> 75[style=dashed];

}

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1([])
f2_in(.(z0, [])) → U1(f18_in(z0), .(z0, []))
f2_in(.(z0, .(z1, z2))) → U2(f38_in(z2, z1, z0), .(z0, .(z1, z2)))
U1(f18_out1(z0), .(z1, [])) → f2_out1(z0)
U1(f18_out2(z0, z1), .(z2, [])) → f2_out1(z1)
U2(f38_out1(z0, z1, z2), .(z3, .(z4, z5))) → f2_out1(z2)
f57_in([], z0) → f57_out1(.(z0, []))
f57_in(.(z0, z1), z2) → U3(f57_in(z1, z2), .(z0, z1), z2)
U3(f57_out1(z0), .(z1, z2), z3) → f57_out1(.(z1, z0))
f42_in([]) → f42_out1([])
f42_in(.(z0, z1)) → U4(f54_in(z1, z0), .(z0, z1))
U4(f54_out1(z0, z1), .(z2, z3)) → f42_out1(z1)
f75_in([], z0) → f75_out1(.(z0, []))
f75_in(.(z0, z1), z2) → U5(f75_in(z1, z2), .(z0, z1), z2)
U5(f75_out1(z0), .(z1, z2), z3) → f75_out1(.(z1, z0))
f20_in(z0) → f20_out1(.(z0, []))
f38_in(z0, z1, z2) → U6(f42_in(z0), z0, z1, z2)
U6(f42_out1(z0), z1, z2, z3) → U7(f43_in(z0, z2, z3), z1, z2, z3, z0)
U7(f43_out1(z0, z1), z2, z3, z4, z5) → f38_out1(z5, z0, z1)
f43_in(z0, z1, z2) → U8(f57_in(z0, z1), z0, z1, z2)
U8(f57_out1(z0), z1, z2, z3) → U9(f75_in(z0, z3), z1, z2, z3, z0)
U9(f75_out1(z0), z1, z2, z3, z4) → f43_out1(z4, z0)
f54_in(z0, z1) → U10(f42_in(z0), z0, z1)
U10(f42_out1(z0), z1, z2) → U11(f57_in(z0, z2), z1, z2, z0)
U11(f57_out1(z0), z1, z2, z3) → f54_out1(z3, z0)
f18_in(z0) → U12(f20_in(z0), f21_in(z0), z0)
U12(f20_out1(z0), z1, z2) → f18_out1(z0)
U12(z0, f21_out1(z1, z2), z3) → f18_out2(z1, z2)

Tuples:

F2_IN(.(z0, [])) → c1(U1'(f18_in(z0), .(z0, [])), F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c2(U2'(f38_in(z2, z1, z0), .(z0, .(z1, z2))), F38_IN(z2, z1, z0))
F57_IN(.(z0, z1), z2) → c7(U3'(f57_in(z1, z2), .(z0, z1), z2), F57_IN(z1, z2))
F42_IN(.(z0, z1)) → c10(U4'(f54_in(z1, z0), .(z0, z1)), F54_IN(z1, z0))
F75_IN(.(z0, z1), z2) → c13(U5'(f75_in(z1, z2), .(z0, z1), z2), F75_IN(z1, z2))
F38_IN(z0, z1, z2) → c16(U6'(f42_in(z0), z0, z1, z2), F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c17(U7'(f43_in(z0, z2, z3), z1, z2, z3, z0), F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c19(U8'(f57_in(z0, z1), z0, z1, z2), F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c20(U9'(f75_in(z0, z3), z1, z2, z3, z0), F75_IN(z0, z3))
F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
U10'(f42_out1(z0), z1, z2) → c23(U11'(f57_in(z0, z2), z1, z2, z0), F57_IN(z0, z2))
F18_IN(z0) → c25(U12'(f20_in(z0), f21_in(z0), z0), F20_IN(z0))

S tuples:

F2_IN(.(z0, [])) → c1(U1'(f18_in(z0), .(z0, [])), F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c2(U2'(f38_in(z2, z1, z0), .(z0, .(z1, z2))), F38_IN(z2, z1, z0))
F57_IN(.(z0, z1), z2) → c7(U3'(f57_in(z1, z2), .(z0, z1), z2), F57_IN(z1, z2))
F42_IN(.(z0, z1)) → c10(U4'(f54_in(z1, z0), .(z0, z1)), F54_IN(z1, z0))
F75_IN(.(z0, z1), z2) → c13(U5'(f75_in(z1, z2), .(z0, z1), z2), F75_IN(z1, z2))
F38_IN(z0, z1, z2) → c16(U6'(f42_in(z0), z0, z1, z2), F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c17(U7'(f43_in(z0, z2, z3), z1, z2, z3, z0), F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c19(U8'(f57_in(z0, z1), z0, z1, z2), F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c20(U9'(f75_in(z0, z3), z1, z2, z3, z0), F75_IN(z0, z3))
F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
U10'(f42_out1(z0), z1, z2) → c23(U11'(f57_in(z0, z2), z1, z2, z0), F57_IN(z0, z2))
F18_IN(z0) → c25(U12'(f20_in(z0), f21_in(z0), z0), F20_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f57_in, U3, f42_in, U4, f75_in, U5, f20_in, f38_in, U6, U7, f43_in, U8, U9, f54_in, U10, U11, f18_in, U12

Defined Pair Symbols:

F2_IN, F57_IN, F42_IN, F75_IN, F38_IN, U6', F43_IN, U8', F54_IN, U10', F18_IN

Compound Symbols:

c1, c2, c7, c10, c13, c16, c17, c19, c20, c22, c23, c25

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

Split RHS of tuples not part of any SCC

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1([])
f2_in(.(z0, [])) → U1(f18_in(z0), .(z0, []))
f2_in(.(z0, .(z1, z2))) → U2(f38_in(z2, z1, z0), .(z0, .(z1, z2)))
U1(f18_out1(z0), .(z1, [])) → f2_out1(z0)
U1(f18_out2(z0, z1), .(z2, [])) → f2_out1(z1)
U2(f38_out1(z0, z1, z2), .(z3, .(z4, z5))) → f2_out1(z2)
f57_in([], z0) → f57_out1(.(z0, []))
f57_in(.(z0, z1), z2) → U3(f57_in(z1, z2), .(z0, z1), z2)
U3(f57_out1(z0), .(z1, z2), z3) → f57_out1(.(z1, z0))
f42_in([]) → f42_out1([])
f42_in(.(z0, z1)) → U4(f54_in(z1, z0), .(z0, z1))
U4(f54_out1(z0, z1), .(z2, z3)) → f42_out1(z1)
f75_in([], z0) → f75_out1(.(z0, []))
f75_in(.(z0, z1), z2) → U5(f75_in(z1, z2), .(z0, z1), z2)
U5(f75_out1(z0), .(z1, z2), z3) → f75_out1(.(z1, z0))
f20_in(z0) → f20_out1(.(z0, []))
f38_in(z0, z1, z2) → U6(f42_in(z0), z0, z1, z2)
U6(f42_out1(z0), z1, z2, z3) → U7(f43_in(z0, z2, z3), z1, z2, z3, z0)
U7(f43_out1(z0, z1), z2, z3, z4, z5) → f38_out1(z5, z0, z1)
f43_in(z0, z1, z2) → U8(f57_in(z0, z1), z0, z1, z2)
U8(f57_out1(z0), z1, z2, z3) → U9(f75_in(z0, z3), z1, z2, z3, z0)
U9(f75_out1(z0), z1, z2, z3, z4) → f43_out1(z4, z0)
f54_in(z0, z1) → U10(f42_in(z0), z0, z1)
U10(f42_out1(z0), z1, z2) → U11(f57_in(z0, z2), z1, z2, z0)
U11(f57_out1(z0), z1, z2, z3) → f54_out1(z3, z0)
f18_in(z0) → U12(f20_in(z0), f21_in(z0), z0)
U12(f20_out1(z0), z1, z2) → f18_out1(z0)
U12(z0, f21_out1(z1, z2), z3) → f18_out2(z1, z2)

Tuples:

F57_IN(.(z0, z1), z2) → c7(U3'(f57_in(z1, z2), .(z0, z1), z2), F57_IN(z1, z2))
F42_IN(.(z0, z1)) → c10(U4'(f54_in(z1, z0), .(z0, z1)), F54_IN(z1, z0))
F75_IN(.(z0, z1), z2) → c13(U5'(f75_in(z1, z2), .(z0, z1), z2), F75_IN(z1, z2))
F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
F2_IN(.(z0, [])) → c(U1'(f18_in(z0), .(z0, [])))
F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(U2'(f38_in(z2, z1, z0), .(z0, .(z1, z2))))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(U7'(f43_in(z0, z2, z3), z1, z2, z3, z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(U9'(f75_in(z0, z3), z1, z2, z3, z0))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
U10'(f42_out1(z0), z1, z2) → c(U11'(f57_in(z0, z2), z1, z2, z0))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
F18_IN(z0) → c(U12'(f20_in(z0), f21_in(z0), z0))
F18_IN(z0) → c(F20_IN(z0))

S tuples:

F57_IN(.(z0, z1), z2) → c7(U3'(f57_in(z1, z2), .(z0, z1), z2), F57_IN(z1, z2))
F42_IN(.(z0, z1)) → c10(U4'(f54_in(z1, z0), .(z0, z1)), F54_IN(z1, z0))
F75_IN(.(z0, z1), z2) → c13(U5'(f75_in(z1, z2), .(z0, z1), z2), F75_IN(z1, z2))
F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
F2_IN(.(z0, [])) → c(U1'(f18_in(z0), .(z0, [])))
F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(U2'(f38_in(z2, z1, z0), .(z0, .(z1, z2))))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(U7'(f43_in(z0, z2, z3), z1, z2, z3, z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(U9'(f75_in(z0, z3), z1, z2, z3, z0))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
U10'(f42_out1(z0), z1, z2) → c(U11'(f57_in(z0, z2), z1, z2, z0))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
F18_IN(z0) → c(U12'(f20_in(z0), f21_in(z0), z0))
F18_IN(z0) → c(F20_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f57_in, U3, f42_in, U4, f75_in, U5, f20_in, f38_in, U6, U7, f43_in, U8, U9, f54_in, U10, U11, f18_in, U12

Defined Pair Symbols:

F57_IN, F42_IN, F75_IN, F54_IN, F2_IN, F38_IN, U6', F43_IN, U8', U10', F18_IN

Compound Symbols:

c7, c10, c13, c22, c

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

Removed 10 trailing tuple parts

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1([])
f2_in(.(z0, [])) → U1(f18_in(z0), .(z0, []))
f2_in(.(z0, .(z1, z2))) → U2(f38_in(z2, z1, z0), .(z0, .(z1, z2)))
U1(f18_out1(z0), .(z1, [])) → f2_out1(z0)
U1(f18_out2(z0, z1), .(z2, [])) → f2_out1(z1)
U2(f38_out1(z0, z1, z2), .(z3, .(z4, z5))) → f2_out1(z2)
f57_in([], z0) → f57_out1(.(z0, []))
f57_in(.(z0, z1), z2) → U3(f57_in(z1, z2), .(z0, z1), z2)
U3(f57_out1(z0), .(z1, z2), z3) → f57_out1(.(z1, z0))
f42_in([]) → f42_out1([])
f42_in(.(z0, z1)) → U4(f54_in(z1, z0), .(z0, z1))
U4(f54_out1(z0, z1), .(z2, z3)) → f42_out1(z1)
f75_in([], z0) → f75_out1(.(z0, []))
f75_in(.(z0, z1), z2) → U5(f75_in(z1, z2), .(z0, z1), z2)
U5(f75_out1(z0), .(z1, z2), z3) → f75_out1(.(z1, z0))
f20_in(z0) → f20_out1(.(z0, []))
f38_in(z0, z1, z2) → U6(f42_in(z0), z0, z1, z2)
U6(f42_out1(z0), z1, z2, z3) → U7(f43_in(z0, z2, z3), z1, z2, z3, z0)
U7(f43_out1(z0, z1), z2, z3, z4, z5) → f38_out1(z5, z0, z1)
f43_in(z0, z1, z2) → U8(f57_in(z0, z1), z0, z1, z2)
U8(f57_out1(z0), z1, z2, z3) → U9(f75_in(z0, z3), z1, z2, z3, z0)
U9(f75_out1(z0), z1, z2, z3, z4) → f43_out1(z4, z0)
f54_in(z0, z1) → U10(f42_in(z0), z0, z1)
U10(f42_out1(z0), z1, z2) → U11(f57_in(z0, z2), z1, z2, z0)
U11(f57_out1(z0), z1, z2, z3) → f54_out1(z3, z0)
f18_in(z0) → U12(f20_in(z0), f21_in(z0), z0)
U12(f20_out1(z0), z1, z2) → f18_out1(z0)
U12(z0, f21_out1(z1, z2), z3) → f18_out2(z1, z2)

Tuples:

F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
F57_IN(.(z0, z1), z2) → c7(F57_IN(z1, z2))
F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))
F75_IN(.(z0, z1), z2) → c13(F75_IN(z1, z2))
F2_IN(.(z0, [])) → c
F2_IN(.(z0, .(z1, z2))) → c
U6'(f42_out1(z0), z1, z2, z3) → c
U8'(f57_out1(z0), z1, z2, z3) → c
U10'(f42_out1(z0), z1, z2) → c
F18_IN(z0) → c

S tuples:

F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
F57_IN(.(z0, z1), z2) → c7(F57_IN(z1, z2))
F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))
F75_IN(.(z0, z1), z2) → c13(F75_IN(z1, z2))
F2_IN(.(z0, [])) → c
F2_IN(.(z0, .(z1, z2))) → c
U6'(f42_out1(z0), z1, z2, z3) → c
U8'(f57_out1(z0), z1, z2, z3) → c
U10'(f42_out1(z0), z1, z2) → c
F18_IN(z0) → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f57_in, U3, f42_in, U4, f75_in, U5, f20_in, f38_in, U6, U7, f43_in, U8, U9, f54_in, U10, U11, f18_in, U12

Defined Pair Symbols:

F54_IN, F2_IN, F38_IN, U6', F43_IN, U8', U10', F57_IN, F42_IN, F75_IN, F18_IN

Compound Symbols:

c22, c, c7, c10, c13, c

(27) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
F2_IN(.(z0, [])) → c
F2_IN(.(z0, .(z1, z2))) → c
U6'(f42_out1(z0), z1, z2, z3) → c
U8'(f57_out1(z0), z1, z2, z3) → c
F18_IN(z0) → c
F18_IN(z0) → c
F18_IN(z0) → c
F18_IN(z0) → c
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
U6'(f42_out1(z0), z1, z2, z3) → c
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
U8'(f57_out1(z0), z1, z2, z3) → c

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1([])
f2_in(.(z0, [])) → U1(f18_in(z0), .(z0, []))
f2_in(.(z0, .(z1, z2))) → U2(f38_in(z2, z1, z0), .(z0, .(z1, z2)))
U1(f18_out1(z0), .(z1, [])) → f2_out1(z0)
U1(f18_out2(z0, z1), .(z2, [])) → f2_out1(z1)
U2(f38_out1(z0, z1, z2), .(z3, .(z4, z5))) → f2_out1(z2)
f57_in([], z0) → f57_out1(.(z0, []))
f57_in(.(z0, z1), z2) → U3(f57_in(z1, z2), .(z0, z1), z2)
U3(f57_out1(z0), .(z1, z2), z3) → f57_out1(.(z1, z0))
f42_in([]) → f42_out1([])
f42_in(.(z0, z1)) → U4(f54_in(z1, z0), .(z0, z1))
U4(f54_out1(z0, z1), .(z2, z3)) → f42_out1(z1)
f75_in([], z0) → f75_out1(.(z0, []))
f75_in(.(z0, z1), z2) → U5(f75_in(z1, z2), .(z0, z1), z2)
U5(f75_out1(z0), .(z1, z2), z3) → f75_out1(.(z1, z0))
f20_in(z0) → f20_out1(.(z0, []))
f38_in(z0, z1, z2) → U6(f42_in(z0), z0, z1, z2)
U6(f42_out1(z0), z1, z2, z3) → U7(f43_in(z0, z2, z3), z1, z2, z3, z0)
U7(f43_out1(z0, z1), z2, z3, z4, z5) → f38_out1(z5, z0, z1)
f43_in(z0, z1, z2) → U8(f57_in(z0, z1), z0, z1, z2)
U8(f57_out1(z0), z1, z2, z3) → U9(f75_in(z0, z3), z1, z2, z3, z0)
U9(f75_out1(z0), z1, z2, z3, z4) → f43_out1(z4, z0)
f54_in(z0, z1) → U10(f42_in(z0), z0, z1)
U10(f42_out1(z0), z1, z2) → U11(f57_in(z0, z2), z1, z2, z0)
U11(f57_out1(z0), z1, z2, z3) → f54_out1(z3, z0)
f18_in(z0) → U12(f20_in(z0), f21_in(z0), z0)
U12(f20_out1(z0), z1, z2) → f18_out1(z0)
U12(z0, f21_out1(z1, z2), z3) → f18_out2(z1, z2)

Tuples:

F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
F57_IN(.(z0, z1), z2) → c7(F57_IN(z1, z2))
F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))
F75_IN(.(z0, z1), z2) → c13(F75_IN(z1, z2))
F2_IN(.(z0, [])) → c
F2_IN(.(z0, .(z1, z2))) → c
U6'(f42_out1(z0), z1, z2, z3) → c
U8'(f57_out1(z0), z1, z2, z3) → c
U10'(f42_out1(z0), z1, z2) → c
F18_IN(z0) → c

S tuples:

F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
F57_IN(.(z0, z1), z2) → c7(F57_IN(z1, z2))
F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))
F75_IN(.(z0, z1), z2) → c13(F75_IN(z1, z2))
U10'(f42_out1(z0), z1, z2) → c

K tuples:

F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
F2_IN(.(z0, [])) → c
F2_IN(.(z0, .(z1, z2))) → c
U6'(f42_out1(z0), z1, z2, z3) → c
U8'(f57_out1(z0), z1, z2, z3) → c
F18_IN(z0) → c

Defined Rule Symbols:

f2_in, U1, U2, f57_in, U3, f42_in, U4, f75_in, U5, f20_in, f38_in, U6, U7, f43_in, U8, U9, f54_in, U10, U11, f18_in, U12

Defined Pair Symbols:

F54_IN, F2_IN, F38_IN, U6', F43_IN, U8', U10', F57_IN, F42_IN, F75_IN, F18_IN

Compound Symbols:

c22, c, c7, c10, c13, c

(29) 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.

F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))

We considered the (Usable) Rules:

f57_in([], z0) → f57_out1(.(z0, []))
f57_in(.(z0, z1), z2) → U3(f57_in(z1, z2), .(z0, z1), z2)
U3(f57_out1(z0), .(z1, z2), z3) → f57_out1(.(z1, z0))
f42_in([]) → f42_out1([])
f42_in(.(z0, z1)) → U4(f54_in(z1, z0), .(z0, z1))
f54_in(z0, z1) → U10(f42_in(z0), z0, z1)
U4(f54_out1(z0, z1), .(z2, z3)) → f42_out1(z1)
U10(f42_out1(z0), z1, z2) → U11(f57_in(z0, z2), z1, z2, z0)
U11(f57_out1(z0), z1, z2, z3) → f54_out1(z3, z0)

And the Tuples:

F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
F57_IN(.(z0, z1), z2) → c7(F57_IN(z1, z2))
F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))
F75_IN(.(z0, z1), z2) → c13(F75_IN(z1, z2))
F2_IN(.(z0, [])) → c
F2_IN(.(z0, .(z1, z2))) → c
U6'(f42_out1(z0), z1, z2, z3) → c
U8'(f57_out1(z0), z1, z2, z3) → c
U10'(f42_out1(z0), z1, z2) → c
F18_IN(z0) → c

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F18_IN(x₁)) = [2]   
POL(F2_IN(x₁)) = [2]x₁   
POL(F38_IN(x₁, x₂, x₃)) = [2] + [2]x₁   
POL(F42_IN(x₁)) = x₁   
POL(F43_IN(x₁, x₂, x₃)) = 0   
POL(F54_IN(x₁, x₂)) = x₁   
POL(F57_IN(x₁, x₂)) = 0   
POL(F75_IN(x₁, x₂)) = 0   
POL(U10(x₁, x₂, x₃)) = 0   
POL(U10'(x₁, x₂, x₃)) = 0   
POL(U11(x₁, x₂, x₃, x₄)) = 0   
POL(U3(x₁, x₂, x₃)) = [2]x₃   
POL(U4(x₁, x₂)) = 0   
POL(U6'(x₁, x₂, x₃, x₄)) = [1]   
POL(U8'(x₁, x₂, x₃, x₄)) = 0   
POL([]) = 0   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c22(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(f42_in(x₁)) = 0   
POL(f42_out1(x₁)) = 0   
POL(f54_in(x₁, x₂)) = 0   
POL(f54_out1(x₁, x₂)) = 0   
POL(f57_in(x₁, x₂)) = [1] + [2]x₂   
POL(f57_out1(x₁)) = 0   

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1([])
f2_in(.(z0, [])) → U1(f18_in(z0), .(z0, []))
f2_in(.(z0, .(z1, z2))) → U2(f38_in(z2, z1, z0), .(z0, .(z1, z2)))
U1(f18_out1(z0), .(z1, [])) → f2_out1(z0)
U1(f18_out2(z0, z1), .(z2, [])) → f2_out1(z1)
U2(f38_out1(z0, z1, z2), .(z3, .(z4, z5))) → f2_out1(z2)
f57_in([], z0) → f57_out1(.(z0, []))
f57_in(.(z0, z1), z2) → U3(f57_in(z1, z2), .(z0, z1), z2)
U3(f57_out1(z0), .(z1, z2), z3) → f57_out1(.(z1, z0))
f42_in([]) → f42_out1([])
f42_in(.(z0, z1)) → U4(f54_in(z1, z0), .(z0, z1))
U4(f54_out1(z0, z1), .(z2, z3)) → f42_out1(z1)
f75_in([], z0) → f75_out1(.(z0, []))
f75_in(.(z0, z1), z2) → U5(f75_in(z1, z2), .(z0, z1), z2)
U5(f75_out1(z0), .(z1, z2), z3) → f75_out1(.(z1, z0))
f20_in(z0) → f20_out1(.(z0, []))
f38_in(z0, z1, z2) → U6(f42_in(z0), z0, z1, z2)
U6(f42_out1(z0), z1, z2, z3) → U7(f43_in(z0, z2, z3), z1, z2, z3, z0)
U7(f43_out1(z0, z1), z2, z3, z4, z5) → f38_out1(z5, z0, z1)
f43_in(z0, z1, z2) → U8(f57_in(z0, z1), z0, z1, z2)
U8(f57_out1(z0), z1, z2, z3) → U9(f75_in(z0, z3), z1, z2, z3, z0)
U9(f75_out1(z0), z1, z2, z3, z4) → f43_out1(z4, z0)
f54_in(z0, z1) → U10(f42_in(z0), z0, z1)
U10(f42_out1(z0), z1, z2) → U11(f57_in(z0, z2), z1, z2, z0)
U11(f57_out1(z0), z1, z2, z3) → f54_out1(z3, z0)
f18_in(z0) → U12(f20_in(z0), f21_in(z0), z0)
U12(f20_out1(z0), z1, z2) → f18_out1(z0)
U12(z0, f21_out1(z1, z2), z3) → f18_out2(z1, z2)

Tuples:

F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
F57_IN(.(z0, z1), z2) → c7(F57_IN(z1, z2))
F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))
F75_IN(.(z0, z1), z2) → c13(F75_IN(z1, z2))
F2_IN(.(z0, [])) → c
F2_IN(.(z0, .(z1, z2))) → c
U6'(f42_out1(z0), z1, z2, z3) → c
U8'(f57_out1(z0), z1, z2, z3) → c
U10'(f42_out1(z0), z1, z2) → c
F18_IN(z0) → c

S tuples:

F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
F57_IN(.(z0, z1), z2) → c7(F57_IN(z1, z2))
F75_IN(.(z0, z1), z2) → c13(F75_IN(z1, z2))
U10'(f42_out1(z0), z1, z2) → c

K tuples:

F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
F2_IN(.(z0, [])) → c
F2_IN(.(z0, .(z1, z2))) → c
U6'(f42_out1(z0), z1, z2, z3) → c
U8'(f57_out1(z0), z1, z2, z3) → c
F18_IN(z0) → c
F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))

Defined Rule Symbols:

f2_in, U1, U2, f57_in, U3, f42_in, U4, f75_in, U5, f20_in, f38_in, U6, U7, f43_in, U8, U9, f54_in, U10, U11, f18_in, U12

Defined Pair Symbols:

F54_IN, F2_IN, F38_IN, U6', F43_IN, U8', U10', F57_IN, F42_IN, F75_IN, F18_IN

Compound Symbols:

c22, c, c7, c10, c13, c

(31) CdtKnowledgeProof (EQUIVALENT transformation)

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

F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
U10'(f42_out1(z0), z1, z2) → c
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))
U10'(f42_out1(z0), z1, z2) → c

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1([])
f2_in(.(z0, [])) → U1(f18_in(z0), .(z0, []))
f2_in(.(z0, .(z1, z2))) → U2(f38_in(z2, z1, z0), .(z0, .(z1, z2)))
U1(f18_out1(z0), .(z1, [])) → f2_out1(z0)
U1(f18_out2(z0, z1), .(z2, [])) → f2_out1(z1)
U2(f38_out1(z0, z1, z2), .(z3, .(z4, z5))) → f2_out1(z2)
f57_in([], z0) → f57_out1(.(z0, []))
f57_in(.(z0, z1), z2) → U3(f57_in(z1, z2), .(z0, z1), z2)
U3(f57_out1(z0), .(z1, z2), z3) → f57_out1(.(z1, z0))
f42_in([]) → f42_out1([])
f42_in(.(z0, z1)) → U4(f54_in(z1, z0), .(z0, z1))
U4(f54_out1(z0, z1), .(z2, z3)) → f42_out1(z1)
f75_in([], z0) → f75_out1(.(z0, []))
f75_in(.(z0, z1), z2) → U5(f75_in(z1, z2), .(z0, z1), z2)
U5(f75_out1(z0), .(z1, z2), z3) → f75_out1(.(z1, z0))
f20_in(z0) → f20_out1(.(z0, []))
f38_in(z0, z1, z2) → U6(f42_in(z0), z0, z1, z2)
U6(f42_out1(z0), z1, z2, z3) → U7(f43_in(z0, z2, z3), z1, z2, z3, z0)
U7(f43_out1(z0, z1), z2, z3, z4, z5) → f38_out1(z5, z0, z1)
f43_in(z0, z1, z2) → U8(f57_in(z0, z1), z0, z1, z2)
U8(f57_out1(z0), z1, z2, z3) → U9(f75_in(z0, z3), z1, z2, z3, z0)
U9(f75_out1(z0), z1, z2, z3, z4) → f43_out1(z4, z0)
f54_in(z0, z1) → U10(f42_in(z0), z0, z1)
U10(f42_out1(z0), z1, z2) → U11(f57_in(z0, z2), z1, z2, z0)
U11(f57_out1(z0), z1, z2, z3) → f54_out1(z3, z0)
f18_in(z0) → U12(f20_in(z0), f21_in(z0), z0)
U12(f20_out1(z0), z1, z2) → f18_out1(z0)
U12(z0, f21_out1(z1, z2), z3) → f18_out2(z1, z2)

Tuples:

F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
F57_IN(.(z0, z1), z2) → c7(F57_IN(z1, z2))
F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))
F75_IN(.(z0, z1), z2) → c13(F75_IN(z1, z2))
F2_IN(.(z0, [])) → c
F2_IN(.(z0, .(z1, z2))) → c
U6'(f42_out1(z0), z1, z2, z3) → c
U8'(f57_out1(z0), z1, z2, z3) → c
U10'(f42_out1(z0), z1, z2) → c
F18_IN(z0) → c

S tuples:

F57_IN(.(z0, z1), z2) → c7(F57_IN(z1, z2))
F75_IN(.(z0, z1), z2) → c13(F75_IN(z1, z2))

K tuples:

F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
F2_IN(.(z0, [])) → c
F2_IN(.(z0, .(z1, z2))) → c
U6'(f42_out1(z0), z1, z2, z3) → c
U8'(f57_out1(z0), z1, z2, z3) → c
F18_IN(z0) → c
F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))
F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
U10'(f42_out1(z0), z1, z2) → c

Defined Rule Symbols:

f2_in, U1, U2, f57_in, U3, f42_in, U4, f75_in, U5, f20_in, f38_in, U6, U7, f43_in, U8, U9, f54_in, U10, U11, f18_in, U12

Defined Pair Symbols:

F54_IN, F2_IN, F38_IN, U6', F43_IN, U8', U10', F57_IN, F42_IN, F75_IN, F18_IN

Compound Symbols:

c22, c, c7, c10, c13, c

(33) 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.

F75_IN(.(z0, z1), z2) → c13(F75_IN(z1, z2))

We considered the (Usable) Rules:

f57_in([], z0) → f57_out1(.(z0, []))
f57_in(.(z0, z1), z2) → U3(f57_in(z1, z2), .(z0, z1), z2)
U3(f57_out1(z0), .(z1, z2), z3) → f57_out1(.(z1, z0))
f42_in([]) → f42_out1([])
f42_in(.(z0, z1)) → U4(f54_in(z1, z0), .(z0, z1))
f54_in(z0, z1) → U10(f42_in(z0), z0, z1)
U4(f54_out1(z0, z1), .(z2, z3)) → f42_out1(z1)
U10(f42_out1(z0), z1, z2) → U11(f57_in(z0, z2), z1, z2, z0)
U11(f57_out1(z0), z1, z2, z3) → f54_out1(z3, z0)

And the Tuples:

F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
F57_IN(.(z0, z1), z2) → c7(F57_IN(z1, z2))
F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))
F75_IN(.(z0, z1), z2) → c13(F75_IN(z1, z2))
F2_IN(.(z0, [])) → c
F2_IN(.(z0, .(z1, z2))) → c
U6'(f42_out1(z0), z1, z2, z3) → c
U8'(f57_out1(z0), z1, z2, z3) → c
U10'(f42_out1(z0), z1, z2) → c
F18_IN(z0) → c

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

POL(.(x₁, x₂)) = [2] + x₂   
POL(F18_IN(x₁)) = 0   
POL(F2_IN(x₁)) = [2]x₁   
POL(F38_IN(x₁, x₂, x₃)) = [3] + [2]x₁   
POL(F42_IN(x₁)) = x₁   
POL(F43_IN(x₁, x₂, x₃)) = [3] + [2]x₁   
POL(F54_IN(x₁, x₂)) = x₁   
POL(F57_IN(x₁, x₂)) = 0   
POL(F75_IN(x₁, x₂)) = [1] + x₁   
POL(U10(x₁, x₂, x₃)) = [1] + x₁   
POL(U10'(x₁, x₂, x₃)) = 0   
POL(U11(x₁, x₂, x₃, x₄)) = x₁   
POL(U3(x₁, x₂, x₃)) = [2] + x₁   
POL(U4(x₁, x₂)) = [1] + x₁   
POL(U6'(x₁, x₂, x₃, x₄)) = [1] + [2]x₁   
POL(U8'(x₁, x₂, x₃, x₄)) = [1] + x₁   
POL([]) = 0   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c22(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(f42_in(x₁)) = [1] + x₁   
POL(f42_out1(x₁)) = [1] + x₁   
POL(f54_in(x₁, x₂)) = [2] + x₁   
POL(f54_out1(x₁, x₂)) = x₂   
POL(f57_in(x₁, x₂)) = [2] + x₁   
POL(f57_out1(x₁)) = x₁   

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1([])
f2_in(.(z0, [])) → U1(f18_in(z0), .(z0, []))
f2_in(.(z0, .(z1, z2))) → U2(f38_in(z2, z1, z0), .(z0, .(z1, z2)))
U1(f18_out1(z0), .(z1, [])) → f2_out1(z0)
U1(f18_out2(z0, z1), .(z2, [])) → f2_out1(z1)
U2(f38_out1(z0, z1, z2), .(z3, .(z4, z5))) → f2_out1(z2)
f57_in([], z0) → f57_out1(.(z0, []))
f57_in(.(z0, z1), z2) → U3(f57_in(z1, z2), .(z0, z1), z2)
U3(f57_out1(z0), .(z1, z2), z3) → f57_out1(.(z1, z0))
f42_in([]) → f42_out1([])
f42_in(.(z0, z1)) → U4(f54_in(z1, z0), .(z0, z1))
U4(f54_out1(z0, z1), .(z2, z3)) → f42_out1(z1)
f75_in([], z0) → f75_out1(.(z0, []))
f75_in(.(z0, z1), z2) → U5(f75_in(z1, z2), .(z0, z1), z2)
U5(f75_out1(z0), .(z1, z2), z3) → f75_out1(.(z1, z0))
f20_in(z0) → f20_out1(.(z0, []))
f38_in(z0, z1, z2) → U6(f42_in(z0), z0, z1, z2)
U6(f42_out1(z0), z1, z2, z3) → U7(f43_in(z0, z2, z3), z1, z2, z3, z0)
U7(f43_out1(z0, z1), z2, z3, z4, z5) → f38_out1(z5, z0, z1)
f43_in(z0, z1, z2) → U8(f57_in(z0, z1), z0, z1, z2)
U8(f57_out1(z0), z1, z2, z3) → U9(f75_in(z0, z3), z1, z2, z3, z0)
U9(f75_out1(z0), z1, z2, z3, z4) → f43_out1(z4, z0)
f54_in(z0, z1) → U10(f42_in(z0), z0, z1)
U10(f42_out1(z0), z1, z2) → U11(f57_in(z0, z2), z1, z2, z0)
U11(f57_out1(z0), z1, z2, z3) → f54_out1(z3, z0)
f18_in(z0) → U12(f20_in(z0), f21_in(z0), z0)
U12(f20_out1(z0), z1, z2) → f18_out1(z0)
U12(z0, f21_out1(z1, z2), z3) → f18_out2(z1, z2)

Tuples:

F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
F57_IN(.(z0, z1), z2) → c7(F57_IN(z1, z2))
F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))
F75_IN(.(z0, z1), z2) → c13(F75_IN(z1, z2))
F2_IN(.(z0, [])) → c
F2_IN(.(z0, .(z1, z2))) → c
U6'(f42_out1(z0), z1, z2, z3) → c
U8'(f57_out1(z0), z1, z2, z3) → c
U10'(f42_out1(z0), z1, z2) → c
F18_IN(z0) → c

S tuples:

F57_IN(.(z0, z1), z2) → c7(F57_IN(z1, z2))

K tuples:

F2_IN(.(z0, [])) → c(F18_IN(z0))
F2_IN(.(z0, .(z1, z2))) → c(F38_IN(z2, z1, z0))
F38_IN(z0, z1, z2) → c(U6'(f42_in(z0), z0, z1, z2))
F38_IN(z0, z1, z2) → c(F42_IN(z0))
U6'(f42_out1(z0), z1, z2, z3) → c(F43_IN(z0, z2, z3))
F43_IN(z0, z1, z2) → c(U8'(f57_in(z0, z1), z0, z1, z2))
F43_IN(z0, z1, z2) → c(F57_IN(z0, z1))
U8'(f57_out1(z0), z1, z2, z3) → c(F75_IN(z0, z3))
F2_IN(.(z0, [])) → c
F2_IN(.(z0, .(z1, z2))) → c
U6'(f42_out1(z0), z1, z2, z3) → c
U8'(f57_out1(z0), z1, z2, z3) → c
F18_IN(z0) → c
F42_IN(.(z0, z1)) → c10(F54_IN(z1, z0))
F54_IN(z0, z1) → c22(U10'(f42_in(z0), z0, z1), F42_IN(z0))
U10'(f42_out1(z0), z1, z2) → c(F57_IN(z0, z2))
U10'(f42_out1(z0), z1, z2) → c
F75_IN(.(z0, z1), z2) → c13(F75_IN(z1, z2))

Defined Rule Symbols:

f2_in, U1, U2, f57_in, U3, f42_in, U4, f75_in, U5, f20_in, f38_in, U6, U7, f43_in, U8, U9, f54_in, U10, U11, f18_in, U12

Defined Pair Symbols:

F54_IN, F2_IN, F38_IN, U6', F43_IN, U8', U10', F57_IN, F42_IN, F75_IN, F18_IN

Compound Symbols:

c22, c, c7, c10, c13, c