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

(0) Obligation:

Clauses:

append1([], Ys, Ys).
append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs).
append2([], Ys, Ys).
append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs).
sublist(X, Y) :- ','(append1(P, X1, Y), append2(X2, X, P)).

Query: sublist(g,g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

append1([], Ys, Ys).
append2([], Ys, Ys).
append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs).
append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs).
sublist(X, Y) :- ','(append1(P, X1, Y), append2(X2, X, P)).

Query: sublist(g,g)

(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: sublist(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: sublist(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(append1(X15, X16, T10), append2(X17, T9, X15))\n\nT9 is ground\nT10 is ground\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: append1(X15, X16, T10)\n\nT10 is ground\nX15 is free\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: append2(X17, T9, T13)\n\nT9 is ground\nT13 is ground\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: append1(X15, X16, T10), SCOPE: 2, CLAUSE: 0 | append1(X15, X16, T10), SCOPE: 2, CLAUSE: 2\n\nT10 is ground\nX15 is free\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true | append1(X15, X16, T18), SCOPE: 2, CLAUSE: 2\n\nT18 is ground\nX15 is free\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: append1(X15, X16, T18), SCOPE: 2, CLAUSE: 2\n\nT18 is ground\nX15 is free\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: append1(X48, X49, T24)\n\nT24 is ground\nX48 is free\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: append2(X17, T9, T13), SCOPE: 3, CLAUSE: 1 | append2(X17, T9, T13), SCOPE: 3, CLAUSE: 3\n\nT9 is ground\nT13 is ground\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true | append2(X17, T30, T30), SCOPE: 3, CLAUSE: 3\n\nT30 is ground\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: append2(X17, T9, T13), SCOPE: 3, CLAUSE: 3\n\nT9 is ground\nT13 is ground\nX17 is free\nX56 is free\nappend2(X17, T9, T13) !=? append2([], X56, X56)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: append2(X17, T30, T30), SCOPE: 3, CLAUSE: 3\n\nT30 is ground\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: append2(X74, .(T38, T39), T39)\n\nT38 is ground\nT39 is ground\nX74 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: append2(X94, T50, T52)\n\nT50 is ground\nT51 is ground\nT52 is ground\nX17 is free\nX56 is free\nX94 is free\nappend2(X17, T50, .(T51, T52)) !=? append2([], X56, X56)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 42 [arrowhead = none ];
42 -> 3;

43 [label="ONLY EVAL with clause\nsublist(X13, X14) :- ','(append1(X15, X16, X14), append2(X17, X13, X15)).\nand substitutionT1 -> T9,\nX13 -> T9,\nT2 -> T10,\nX14 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 43 [arrowhead = none ];
43 -> 7;

44 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 44 [arrowhead = none ];
44 -> 12;

45 [label="SPLIT 2\nnew knowledge:\nT13 is ground\nT14 is ground\nT10 is ground\nreplacements:X15 -> T13,\nX16 -> T14", fontsize=14, style = filled, fillcolor = violet];
7 -> 45 [arrowhead = none ];
45 -> 13;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 46 [arrowhead = none ];
46 -> 16;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 47 [arrowhead = none ];
47 -> 26;

48 [label="ONLY EVAL with clause\nappend1([], X27, X27).\nand substitutionX15 -> [],\nX16 -> T18,\nX27 -> T18,\nT10 -> T18,\nX28 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 48 [arrowhead = none ];
48 -> 19;

49 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 49 [arrowhead = none ];
49 -> 20;

50 [label="EVAL with clause\nappend1(.(X43, X44), X45, .(X43, X46)) :- append1(X44, X45, X46).\nand substitutionX43 -> T23,\nX44 -> X48,\nX15 -> .(T23, X48),\nX16 -> X49,\nX45 -> X49,\nX47 -> T23,\nX46 -> T24,\nT18 -> .(T23, T24)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 50 [arrowhead = none ];
50 -> 22;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 51 [arrowhead = none ];
51 -> 23;

52 [label="INSTANCE with matching:\nX15 -> X48\nX16 -> X49\nT10 -> T24", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 52 [arrowhead = none , style=dashed];
52 -> 12[style=dashed];

53 [label="EVAL with clause\nappend2([], X56, X56).\nand substitutionX17 -> [],\nT9 -> T30,\nX56 -> T30,\nT13 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 53 [arrowhead = none ];
53 -> 29;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 54 [arrowhead = none ];
54 -> 30;

55 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 55 [arrowhead = none ];
55 -> 31;

56 [label="EVAL with clause\nappend2(.(X89, X90), X91, .(X89, X92)) :- append2(X90, X91, X92).\nand substitutionX89 -> T51,\nX90 -> X94,\nX17 -> .(T51, X94),\nT9 -> T50,\nX91 -> T50,\nX93 -> T51,\nX92 -> T52,\nT13 -> .(T51, T52)", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 56 [arrowhead = none ];
56 -> 39;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 57 [arrowhead = none ];
57 -> 41;

58 [label="EVAL with clause\nappend2(.(X69, X70), X71, .(X69, X72)) :- append2(X70, X71, X72).\nand substitutionX69 -> T38,\nX70 -> X74,\nX17 -> .(T38, X74),\nT30 -> .(T38, T39),\nX71 -> .(T38, T39),\nX73 -> T38,\nX72 -> T39,\nT37 -> .(T38, T39)", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 58 [arrowhead = none ];
58 -> 35;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 59 [arrowhead = none ];
59 -> 36;

60 [label="INSTANCE with matching:\nX17 -> X74\nT9 -> .(T38, T39)\nT13 -> T39", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 60 [arrowhead = none , style=dashed];
60 -> 13[style=dashed];

61 [label="INSTANCE with matching:\nX17 -> X94\nT9 -> T50\nT13 -> T52", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 61 [arrowhead = none , style=dashed];
61 -> 13[style=dashed];

}

(4) Obligation:

Complex Complexity Dependency Tuples Problem
MAX

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))
F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c6(U10'(f13_in(z3, z0), z2, z3, z0, z1), F13_IN(z3, z0))

S tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))
F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c6(U10'(f13_in(z3, z0), z2, z3, z0, z1), F13_IN(z3, z0))

S tuples:

F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))
F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c6(U10'(f13_in(z3, z0), z2, z3, z0, z1), F13_IN(z3, z0))

S tuples:

F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c6(U10'(f13_in(z3, z0), z2, z3, z0, z1), F13_IN(z3, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

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

Took the maximum complexity of the problems.

(6) Complex Obligation (MAX)

(7) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))
F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c6(U10'(f13_in(z3, z0), z2, z3, z0, z1), F13_IN(z3, z0))

S tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

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

Split RHS of tuples not part of any SCC

(9) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))
F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(U1'(f7_in(z1, z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(U10'(f13_in(z3, z0), z2, z3, z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))

S tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))
F2_IN(z0, z1) → c7(U1'(f7_in(z1, z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, U2', F12_IN, U5', F13_IN, F2_IN, U9'

Compound Symbols:

c2, c3, c1, c4, c2, c5, c7

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

Removed 7 trailing tuple parts

(11) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

S tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F2_IN(z0, z1) → c7

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(12) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F2_IN(z0, z1) → c7
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U9'(f12_out1(z0, z1), z2, z3) → c7
U5'(f12_out1(z0, z1), z2, z3) → c4

Now S is empty

(13) BOUNDS(O(1), O(1))

(14) Obligation:

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))
F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c6(U10'(f13_in(z3, z0), z2, z3, z0, z1), F13_IN(z3, z0))

S tuples:

F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))
F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c6(U10'(f13_in(z3, z0), z2, z3, z0, z1), F13_IN(z3, z0))

S tuples:

F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c6(U10'(f13_in(z3, z0), z2, z3, z0, z1), F13_IN(z3, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

(15) MultiplicationProof (BOTH BOUNDS(ID, ID) transformation)

Multiplied the complexity of the problems.

(16) Complex Obligation (MULT)

(17) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))
F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c6(U10'(f13_in(z3, z0), z2, z3, z0, z1), F13_IN(z3, z0))

S tuples:

F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

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

Split RHS of tuples not part of any SCC

(19) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))
F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(U1'(f7_in(z1, z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(U10'(f13_in(z3, z0), z2, z3, z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))

S tuples:

F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, U2', F12_IN, U5', F13_IN, F2_IN, U9'

Compound Symbols:

c2, c3, c1, c4, c2, c5, c7

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

Removed 7 trailing tuple parts

(21) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

S tuples:

F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(22) 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.

U5'(f12_out1(z0, z1), z2, z3) → c4

We considered the (Usable) Rules:

f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)

And the Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

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

POL(.(x₁, x₂)) = 0   
POL(F12_IN(x₁)) = [2]   
POL(F13_IN(x₁, x₂)) = [1] + x₁   
POL(F2_IN(x₁, x₂)) = [2] + [2]x₁ + [3]x₂   
POL(F7_IN(x₁, x₂)) = [2] + [3]x₁ + [2]x₂   
POL(U2'(x₁, x₂, x₃)) = [2] + x₂   
POL(U4(x₁, x₂)) = 0   
POL(U5'(x₁, x₂, x₃)) = [2] + [3]x₂ + [2]x₃   
POL(U9'(x₁, x₂, x₃)) = [2] + [2]x₂ + x₃   
POL([]) = 0   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3) = 0   
POL(c4) = 0   
POL(c5(x₁)) = x₁   
POL(c7) = 0   
POL(c7(x₁)) = x₁   
POL(f12_in(x₁)) = 0   
POL(f12_out1(x₁, x₂)) = 0   

(23) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

S tuples:

F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
F12_IN(.(z0, z1)) → c1(F12_IN(z1))

K tuples:

U5'(f12_out1(z0, z1), z2, z3) → c4

Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(24) 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(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))

We considered the (Usable) Rules:

f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)

And the Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

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

POL(.(x₁, x₂)) = [3] + x₂   
POL(F12_IN(x₁)) = 0   
POL(F13_IN(x₁, x₂)) = 0   
POL(F2_IN(x₁, x₂)) = [3] + [2]x₁ + [3]x₂   
POL(F7_IN(x₁, x₂)) = [2] + [3]x₁ + [2]x₂   
POL(U2'(x₁, x₂, x₃)) = [1] + [3]x₂ + [2]x₃   
POL(U4(x₁, x₂)) = [2]   
POL(U5'(x₁, x₂, x₃)) = x₃   
POL(U9'(x₁, x₂, x₃)) = 0   
POL([]) = 0   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3) = 0   
POL(c4) = 0   
POL(c5(x₁)) = x₁   
POL(c7) = 0   
POL(c7(x₁)) = x₁   
POL(f12_in(x₁)) = [3] + [2]x₁   
POL(f12_out1(x₁, x₂)) = [2]   

(25) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

S tuples:

F12_IN(.(z0, z1)) → c1(F12_IN(z1))

K tuples:

U5'(f12_out1(z0, z1), z2, z3) → c4
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))

Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(26) 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.

F12_IN(.(z0, z1)) → c1(F12_IN(z1))

We considered the (Usable) Rules:

f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)

And the Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F12_IN(x₁)) = [1] + [2]x₁   
POL(F13_IN(x₁, x₂)) = x₁   
POL(F2_IN(x₁, x₂)) = [1] + [2]x₁ + [3]x₂   
POL(F7_IN(x₁, x₂)) = [1] + [2]x₁ + [2]x₂   
POL(U2'(x₁, x₂, x₃)) = [2]x₃   
POL(U4(x₁, x₂)) = 0   
POL(U5'(x₁, x₂, x₃)) = x₂ + [2]x₃   
POL(U9'(x₁, x₂, x₃)) = [1] + x₃   
POL([]) = 0   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3) = 0   
POL(c4) = 0   
POL(c5(x₁)) = x₁   
POL(c7) = 0   
POL(c7(x₁)) = x₁   
POL(f12_in(x₁)) = 0   
POL(f12_out1(x₁, x₂)) = 0   

(27) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

S tuples:none
K tuples:

U5'(f12_out1(z0, z1), z2, z3) → c4
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
F12_IN(.(z0, z1)) → c1(F12_IN(z1))

Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(28) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(29) BOUNDS(O(1), O(1))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))
F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c6(U10'(f13_in(z3, z0), z2, z3, z0, z1), F13_IN(z3, z0))

S tuples:

F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c6(U10'(f13_in(z3, z0), z2, z3, z0, z1), F13_IN(z3, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

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

Split RHS of tuples not part of any SCC

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0, z1), z2, z3) → c3(U3'(f13_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4(U6'(f13_in(z3, z0), z2, z3, z0, z1))
F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(U1'(f7_in(z1, z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(U10'(f13_in(z3, z0), z2, z3, z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))

S tuples:

F13_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c7(U10'(f13_in(z3, z0), z2, z3, z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, U2', F12_IN, U5', F13_IN, F2_IN, U9'

Compound Symbols:

c2, c3, c1, c4, c2, c5, c7

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

Removed 7 trailing tuple parts

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

S tuples:

F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
U9'(f12_out1(z0, z1), z2, z3) → c7

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(35) 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(z0, z1) → c5(U9'(f12_in(z0), z0, z1))

We considered the (Usable) Rules:

f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)

And the Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

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

POL(.(x₁, x₂)) = 0   
POL(F12_IN(x₁)) = [2]   
POL(F13_IN(x₁, x₂)) = x₁   
POL(F2_IN(x₁, x₂)) = [3] + [3]x₁ + [3]x₂   
POL(F7_IN(x₁, x₂)) = [2] + [2]x₂   
POL(U2'(x₁, x₂, x₃)) = [2] + [2]x₃   
POL(U4(x₁, x₂)) = 0   
POL(U5'(x₁, x₂, x₃)) = [1] + x₃   
POL(U9'(x₁, x₂, x₃)) = x₃   
POL([]) = 0   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3) = 0   
POL(c4) = 0   
POL(c5(x₁)) = x₁   
POL(c7) = 0   
POL(c7(x₁)) = x₁   
POL(f12_in(x₁)) = 0   
POL(f12_out1(x₁, x₂)) = 0   

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

S tuples:

U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
U9'(f12_out1(z0, z1), z2, z3) → c7

K tuples:

F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))

Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(37) CdtKnowledgeProof (EQUIVALENT transformation)

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

U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U9'(f12_out1(z0, z1), z2, z3) → c7

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

S tuples:

F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))

K tuples:

F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U9'(f12_out1(z0, z1), z2, z3) → c7

Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(39) 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.

F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))

We considered the (Usable) Rules:

f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)

And the Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F12_IN(x₁)) = [1]   
POL(F13_IN(x₁, x₂)) = x₁ + [2]x₂   
POL(F2_IN(x₁, x₂)) = [3] + [2]x₁ + [2]x₂   
POL(F7_IN(x₁, x₂)) = [3] + [2]x₁ + [2]x₂   
POL(U2'(x₁, x₂, x₃)) = [3] + [2]x₃   
POL(U4(x₁, x₂)) = [1] + x₁   
POL(U5'(x₁, x₂, x₃)) = [2] + x₂ + x₃   
POL(U9'(x₁, x₂, x₃)) = [2]x₁ + x₃   
POL([]) = [1]   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3) = 0   
POL(c4) = 0   
POL(c5(x₁)) = x₁   
POL(c7) = 0   
POL(c7(x₁)) = x₁   
POL(f12_in(x₁)) = [1] + x₁   
POL(f12_out1(x₁, x₂)) = x₁   

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0, z1), z2, z3) → U3(f13_in(z3, z0), z2, z3, z0, z1)
U3(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U5(f12_out1(z0, z1), z2, z3) → U6(f13_in(z3, z0), z2, z3, z0, z1)
U6(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f13_in(z0, z0) → f13_out1([])
f13_in(.(z0, z1), .(z0, z1)) → U7(f13_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1(z0), .(z1, z2), .(z1, z2)) → f13_out1(.(z1, z0))
U8(f13_out1(z0), z1, .(z2, z3)) → f13_out1(.(z2, z0))
U9(f12_out1(z0, z1), z2, z3) → U10(f13_in(z3, z0), z2, z3, z0, z1)
U10(f13_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U2'(f12_out1(z0, z1), z2, z3) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2, z3) → c4
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0, z1), z2, z3) → c7

S tuples:none
K tuples:

F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0, z1), z2, z3) → c7(F13_IN(z3, z0))
U9'(f12_out1(z0, z1), z2, z3) → c7
F13_IN(.(z0, z1), .(z0, z1)) → c1(F13_IN(.(z0, z1), z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))

Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(41) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(42) BOUNDS(O(1), O(1))

(43) 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: sublist(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: sublist(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(append1(X7, X8, T6), append2(X9, T5, X7))\n\nT5 is ground\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(append1(X7, X8, T6), append2(X9, T5, X7)), SCOPE: 2, CLAUSE: 0 | ','(append1(X7, X8, T6), append2(X9, T5, X7)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: append2(X9, T5, []) | ','(append1(X7, X8, T11), append2(X9, T5, X7)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT11 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: append2(X9, T5, [])\n\nT5 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(append1(X7, X8, T11), append2(X9, T5, X7)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT11 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: append2(X9, T5, []), SCOPE: 3, CLAUSE: 1 | append2(X9, T5, []), SCOPE: 3, CLAUSE: 3\n\nT5 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true | append2(X9, [], []), SCOPE: 3, CLAUSE: 3\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: append2(X9, T5, []), SCOPE: 3, CLAUSE: 3\n\nT5 is ground\nX9 is free\nX24 is free\nappend2(X9, T5, []) !=? append2([], X24, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: append2(X9, [], []), SCOPE: 3, CLAUSE: 3\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(append1(X56, X57, T25), append2(X9, T5, .(T24, X56)))\n\nT5 is ground\nT24 is ground\nT25 is ground\nX9 is free\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: append1(X56, X57, T25)\n\nT25 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: append2(X9, T5, .(T24, T33))\n\nT5 is ground\nT24 is ground\nT33 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: append1(X56, X57, T25), SCOPE: 4, CLAUSE: 0 | append1(X56, X57, T25), SCOPE: 4, CLAUSE: 2\n\nT25 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true | append1(X56, X57, T38), SCOPE: 4, CLAUSE: 2\n\nT38 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: append1(X56, X57, T38), SCOPE: 4, CLAUSE: 2\n\nT38 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: append1(X89, X90, T44)\n\nT44 is ground\nX89 is free\nX90 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: append2(X9, T5, .(T24, T33)), SCOPE: 5, CLAUSE: 1 | append2(X9, T5, .(T24, T33)), SCOPE: 5, CLAUSE: 3\n\nT5 is ground\nT24 is ground\nT33 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true | append2(X9, .(T59, T60), .(T59, T60)), SCOPE: 5, CLAUSE: 3\n\nT59 is ground\nT60 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: append2(X9, T5, .(T24, T33)), SCOPE: 5, CLAUSE: 3\n\nT5 is ground\nT24 is ground\nT33 is ground\nX9 is free\nX97 is free\nappend2(X9, T5, .(T24, T33)) !=? append2([], X97, X97)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: append2(X9, .(T59, T60), .(T59, T60)), SCOPE: 5, CLAUSE: 3\n\nT59 is ground\nT60 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: append2(X117, .(T69, T70), T70)\n\nT69 is ground\nT70 is ground\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: append2(X117, .(T69, T70), T70), SCOPE: 6, CLAUSE: 1 | append2(X117, .(T69, T70), T70), SCOPE: 6, CLAUSE: 3\n\nT69 is ground\nT70 is ground\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: append2(X117, .(T69, T70), T70), SCOPE: 6, CLAUSE: 3\n\nT69 is ground\nT70 is ground\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: append2(X138, .(T85, .(T87, T88)), T88)\n\nT85 is ground\nT87 is ground\nT88 is ground\nX138 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: append2(X138, .(T85, .(T87, T88)), T88), SCOPE: 7, CLAUSE: 1 | append2(X138, .(T85, .(T87, T88)), T88), SCOPE: 7, CLAUSE: 3\n\nT85 is ground\nT87 is ground\nT88 is ground\nX138 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: append2(X138, .(T85, .(T87, T88)), T88), SCOPE: 7, CLAUSE: 3\n\nT85 is ground\nT87 is ground\nT88 is ground\nX138 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: append2(X159, .(T108, .(T109, .(T111, T112))), T112)\n\nT108 is ground\nT109 is ground\nT111 is ground\nT112 is ground\nX159 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: append2(X159, .(T108, .(T109, .(T111, T112))), T112), SCOPE: 8, CLAUSE: 1 | append2(X159, .(T108, .(T109, .(T111, T112))), T112), SCOPE: 8, CLAUSE: 3\n\nT108 is ground\nT109 is ground\nT111 is ground\nT112 is ground\nX159 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: append2(X159, .(T108, .(T109, .(T111, T112))), T112), SCOPE: 8, CLAUSE: 3\n\nT108 is ground\nT109 is ground\nT111 is ground\nT112 is ground\nX159 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: append2(X180, .(T137, .(T138, .(T139, .(T141, T142)))), T142)\n\nT137 is ground\nT138 is ground\nT139 is ground\nT141 is ground\nT142 is ground\nX180 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: append2(X180, .(T137, .(T138, .(T139, .(T141, T142)))), T142), SCOPE: 9, CLAUSE: 1 | append2(X180, .(T137, .(T138, .(T139, .(T141, T142)))), T142), SCOPE: 9, CLAUSE: 3\n\nT137 is ground\nT138 is ground\nT139 is ground\nT141 is ground\nT142 is ground\nX180 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: append2(X180, .(T137, .(T138, .(T139, .(T141, T142)))), T142), SCOPE: 9, CLAUSE: 3\n\nT137 is ground\nT138 is ground\nT139 is ground\nT141 is ground\nT142 is ground\nX180 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: append2(X201, .(T172, .(T173, .(T174, .(T175, .(T177, T178))))), T178)\n\nT172 is ground\nT173 is ground\nT174 is ground\nT175 is ground\nT177 is ground\nT178 is ground\nX201 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: append2(X201, .(T172, .(T173, .(T174, .(T175, .(T177, T178))))), T178), SCOPE: 10, CLAUSE: 1 | append2(X201, .(T172, .(T173, .(T174, .(T175, .(T177, T178))))), T178), SCOPE: 10, CLAUSE: 3\n\nT172 is ground\nT173 is ground\nT174 is ground\nT175 is ground\nT177 is ground\nT178 is ground\nX201 is free", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: append2(X201, .(T172, .(T173, .(T174, .(T175, .(T177, T178))))), T178), SCOPE: 10, CLAUSE: 3\n\nT172 is ground\nT173 is ground\nT174 is ground\nT175 is ground\nT177 is ground\nT178 is ground\nX201 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: append2(X222, .(T213, .(T214, .(T215, .(T216, .(T217, .(T219, T220)))))), T220)\n\nT213 is ground\nT214 is ground\nT215 is ground\nT216 is ground\nT217 is ground\nT219 is ground\nT220 is ground\nX222 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: append2(X222, .(T213, .(T214, .(T215, .(T216, .(T217, .(T219, T220)))))), T220), SCOPE: 11, CLAUSE: 1 | append2(X222, .(T213, .(T214, .(T215, .(T216, .(T217, .(T219, T220)))))), T220), SCOPE: 11, CLAUSE: 3\n\nT213 is ground\nT214 is ground\nT215 is ground\nT216 is ground\nT217 is ground\nT219 is ground\nT220 is ground\nX222 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: append2(X222, .(T213, .(T214, .(T215, .(T216, .(T217, .(T219, T220)))))), T220), SCOPE: 11, CLAUSE: 3\n\nT213 is ground\nT214 is ground\nT215 is ground\nT216 is ground\nT217 is ground\nT219 is ground\nT220 is ground\nX222 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: append2(X243, .(T260, .(T261, .(T262, .(T263, .(T264, .(T265, .(T267, T268))))))), T268)\n\nT260 is ground\nT261 is ground\nT262 is ground\nT263 is ground\nT264 is ground\nT265 is ground\nT267 is ground\nT268 is ground\nX243 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: append2(X243, .(T260, .(T261, .(T262, .(T263, .(T264, .(T265, .(T267, T268))))))), T268), SCOPE: 12, CLAUSE: 1 | append2(X243, .(T260, .(T261, .(T262, .(T263, .(T264, .(T265, .(T267, T268))))))), T268), SCOPE: 12, CLAUSE: 3\n\nT260 is ground\nT261 is ground\nT262 is ground\nT263 is ground\nT264 is ground\nT265 is ground\nT267 is ground\nT268 is ground\nX243 is free", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: append2(X243, .(T260, .(T261, .(T262, .(T263, .(T264, .(T265, .(T267, T268))))))), T268), SCOPE: 12, CLAUSE: 3\n\nT260 is ground\nT261 is ground\nT262 is ground\nT263 is ground\nT264 is ground\nT265 is ground\nT267 is ground\nT268 is ground\nX243 is free", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: append2(X264, .(T313, .(T314, .(T315, .(T316, .(T317, .(T318, .(T319, .(T321, T322)))))))), T322)\n\nT313 is ground\nT314 is ground\nT315 is ground\nT316 is ground\nT317 is ground\nT318 is ground\nT319 is ground\nT321 is ground\nT322 is ground\nX264 is free", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: append2(X264, .(T313, T341), T322)\n\nT313 is ground\nT322 is ground\nT341 is ground\nX264 is free", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: append2(X264, .(T313, T341), T322), SCOPE: 13, CLAUSE: 1 | append2(X264, .(T313, T341), T322), SCOPE: 13, CLAUSE: 3\n\nT313 is ground\nT322 is ground\nT341 is ground\nX264 is free", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: true | append2(X264, .(T350, T351), .(T350, T351)), SCOPE: 13, CLAUSE: 3\n\nT350 is ground\nT351 is ground\nX264 is free", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: append2(X264, .(T313, T341), T322), SCOPE: 13, CLAUSE: 3\n\nT313 is ground\nT322 is ground\nT341 is ground\nX264 is free\nX271 is free\nappend2(X264, .(T313, T341), T322) !=? append2([], X271, X271)", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: append2(X264, .(T350, T351), .(T350, T351)), SCOPE: 13, CLAUSE: 3\n\nT350 is ground\nT351 is ground\nX264 is free", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: append2(X291, .(T360, T361), T361)\n\nT360 is ground\nT361 is ground\nX291 is free", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: append2(X311, .(T374, T375), T377)\n\nT374 is ground\nT375 is ground\nT376 is ground\nT377 is ground\nX264 is free\nX271 is free\nX311 is free\nappend2(X264, .(T374, T375), .(T376, T377)) !=? append2([], X271, X271)", fontsize=16, style=filled, fillcolor=lightyellow];
147 [label="147: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
167 [label="167: append2(X333, T390, T392)\n\nT390 is ground\nT391 is ground\nT392 is ground\nX9 is free\nX97 is free\nX333 is free\nappend2(X9, T390, .(T391, T392)) !=? append2([], X97, X97)", fontsize=16, style=filled, fillcolor=lightyellow];
168 [label="168: append2(X333, T390, T392), SCOPE: 14, CLAUSE: 1 | append2(X333, T390, T392), SCOPE: 14, CLAUSE: 3\n\nT390 is ground\nT391 is ground\nT392 is ground\nX9 is free\nX97 is free\nX333 is free\nappend2(X9, T390, .(T391, T392)) !=? append2([], X97, X97)", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="169: true | append2(X333, T397, T397), SCOPE: 14, CLAUSE: 3\n\nT397 is ground\nX333 is free", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="170: append2(X333, T390, T392), SCOPE: 14, CLAUSE: 3\n\nT390 is ground\nT391 is ground\nT392 is ground\nX9 is free\nX97 is free\nX333 is free\nX338 is free\nappend2(X9, T390, .(T391, T392)) !=? append2([], X97, X97)\nappend2(X333, T390, T392) !=? append2([], X338, X338)", fontsize=16, style=filled, fillcolor=lightyellow];
171 [label="171: append2(X333, T397, T397), SCOPE: 14, CLAUSE: 3\n\nT397 is ground\nX333 is free", fontsize=16, style=filled, fillcolor=lightyellow];
185 [label="185: append2(X356, .(T405, T406), T406)\n\nT405 is ground\nT406 is ground\nX356 is free", fontsize=16, style=filled, fillcolor=lightyellow];
186 [label="186: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
187 [label="187: append2(X376, T417, T419)\n\nT391 is ground\nT417 is ground\nT418 is ground\nT419 is ground\nX9 is free\nX97 is free\nX333 is free\nX338 is free\nX376 is free\nappend2(X9, T417, .(T391, .(T418, T419))) !=? append2([], X97, X97)\nappend2(X333, T417, .(T418, T419)) !=? append2([], X338, X338)", fontsize=16, style=filled, fillcolor=lightyellow];
188 [label="188: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
189 [label="189: append2(X376, T417, T419), SCOPE: 15, CLAUSE: 1 | append2(X376, T417, T419), SCOPE: 15, CLAUSE: 3\n\nT391 is ground\nT417 is ground\nT418 is ground\nT419 is ground\nX9 is free\nX97 is free\nX333 is free\nX338 is free\nX376 is free\nappend2(X9, T417, .(T391, .(T418, T419))) !=? append2([], X97, X97)\nappend2(X333, T417, .(T418, T419)) !=? append2([], X338, X338)", fontsize=16, style=filled, fillcolor=lightyellow];
190 [label="190: true | append2(X376, T424, T424), SCOPE: 15, CLAUSE: 3\n\nT424 is ground\nX376 is free", fontsize=16, style=filled, fillcolor=lightyellow];
191 [label="191: append2(X376, T417, T419), SCOPE: 15, CLAUSE: 3\n\nT391 is ground\nT417 is ground\nT418 is ground\nT419 is ground\nX9 is free\nX97 is free\nX333 is free\nX338 is free\nX376 is free\nX381 is free\nappend2(X9, T417, .(T391, .(T418, T419))) !=? append2([], X97, X97)\nappend2(X333, T417, .(T418, T419)) !=? append2([], X338, X338)\nappend2(X376, T417, T419) !=? append2([], X381, X381)", fontsize=16, style=filled, fillcolor=lightyellow];
192 [label="192: append2(X376, T424, T424), SCOPE: 15, CLAUSE: 3\n\nT424 is ground\nX376 is free", fontsize=16, style=filled, fillcolor=lightyellow];
193 [label="193: append2(X399, .(T432, T433), T433)\n\nT432 is ground\nT433 is ground\nX399 is free", fontsize=16, style=filled, fillcolor=lightyellow];
194 [label="194: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
197 [label="197: append2(X419, T444, T446)\n\nT391 is ground\nT418 is ground\nT444 is ground\nT445 is ground\nT446 is ground\nX9 is free\nX97 is free\nX333 is free\nX338 is free\nX376 is free\nX381 is free\nX419 is free\nappend2(X9, T444, .(T391, .(T418, .(T445, T446)))) !=? append2([], X97, X97)\nappend2(X333, T444, .(T418, .(T445, T446))) !=? append2([], X338, X338)\nappend2(X376, T444, .(T445, T446)) !=? append2([], X381, X381)", fontsize=16, style=filled, fillcolor=lightyellow];
199 [label="199: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
200 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 200 [arrowhead = none ];
200 -> 4;

201 [label="ONLY EVAL with clause\nsublist(X5, X6) :- ','(append1(X7, X8, X6), append2(X9, X5, X7)).\nand substitutionT1 -> T5,\nX5 -> T5,\nT2 -> T6,\nX6 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 201 [arrowhead = none ];
201 -> 5;

202 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 202 [arrowhead = none ];
202 -> 6;

203 [label="ONLY EVAL with clause\nappend1([], X16, X16).\nand substitutionX7 -> [],\nX8 -> T11,\nX16 -> T11,\nT6 -> T11,\nX17 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 203 [arrowhead = none ];
203 -> 8;

204 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 204 [arrowhead = none ];
204 -> 9;

205 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 205 [arrowhead = none ];
205 -> 10;

206 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 206 [arrowhead = none ];
206 -> 11;

207 [label="EVAL with clause\nappend1(.(X51, X52), X53, .(X51, X54)) :- append1(X52, X53, X54).\nand substitutionX51 -> T24,\nX52 -> X56,\nX7 -> .(T24, X56),\nX8 -> X57,\nX53 -> X57,\nX55 -> T24,\nX54 -> T25,\nT11 -> .(T24, T25)", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 207 [arrowhead = none ];
207 -> 24;

208 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 208 [arrowhead = none ];
208 -> 25;

209 [label="EVAL with clause\nappend2([], X24, X24).\nand substitutionX9 -> [],\nT5 -> [],\nX24 -> [],\nT18 -> []", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 209 [arrowhead = none ];
209 -> 14;

210 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 210 [arrowhead = none ];
210 -> 15;

211 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 211 [arrowhead = none ];
211 -> 17;

212 [label="BACKTRACK\nfor clause: append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 212 [arrowhead = none ];
212 -> 21;

213 [label="BACKTRACK\nfor clause: append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 213 [arrowhead = none ];
213 -> 18;

214 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
24 -> 214 [arrowhead = none ];
214 -> 27;

215 [label="SPLIT 2\nnew knowledge:\nT33 is ground\nT34 is ground\nT25 is ground\nreplacements:X56 -> T33,\nX57 -> T34", fontsize=14, style = filled, fillcolor = violet];
24 -> 215 [arrowhead = none ];
215 -> 28;

216 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 216 [arrowhead = none ];
216 -> 32;

217 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 217 [arrowhead = none ];
217 -> 40;

218 [label="ONLY EVAL with clause\nappend1([], X68, X68).\nand substitutionX56 -> [],\nX57 -> T38,\nX68 -> T38,\nT25 -> T38,\nX69 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 218 [arrowhead = none ];
218 -> 33;

219 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 219 [arrowhead = none ];
219 -> 34;

220 [label="EVAL with clause\nappend1(.(X84, X85), X86, .(X84, X87)) :- append1(X85, X86, X87).\nand substitutionX84 -> T43,\nX85 -> X89,\nX56 -> .(T43, X89),\nX57 -> X90,\nX86 -> X90,\nX88 -> T43,\nX87 -> T44,\nT38 -> .(T43, T44)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 220 [arrowhead = none ];
220 -> 37;

221 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 221 [arrowhead = none ];
221 -> 38;

222 [label="INSTANCE with matching:\nX56 -> X89\nX57 -> X90\nT25 -> T44", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 222 [arrowhead = none , style=dashed];
222 -> 27[style=dashed];

223 [label="EVAL with clause\nappend2([], X97, X97).\nand substitutionX9 -> [],\nT5 -> .(T59, T60),\nX97 -> .(T59, T60),\nT24 -> T59,\nT33 -> T60,\nT58 -> .(T59, T60)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 223 [arrowhead = none ];
223 -> 42;

224 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 224 [arrowhead = none ];
224 -> 43;

225 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 225 [arrowhead = none ];
225 -> 44;

226 [label="ONLY EVAL with clause\nappend2(.(X328, X329), X330, .(X328, X331)) :- append2(X329, X330, X331).\nand substitutionX328 -> T391,\nX329 -> X333,\nX9 -> .(T391, X333),\nT5 -> T390,\nX330 -> T390,\nT24 -> T391,\nX332 -> T391,\nT33 -> T392,\nX331 -> T392", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 226 [arrowhead = none ];
226 -> 167;

227 [label="ONLY EVAL with clause\nappend2(.(X112, X113), X114, .(X112, X115)) :- append2(X113, X114, X115).\nand substitutionX112 -> T69,\nX113 -> X117,\nX9 -> .(T69, X117),\nT59 -> T69,\nT60 -> T70,\nX114 -> .(T69, T70),\nX116 -> T69,\nX115 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 227 [arrowhead = none ];
227 -> 45;

228 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 228 [arrowhead = none ];
228 -> 46;

229 [label="BACKTRACK\nfor clause: append2([], Ys, Ys)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 229 [arrowhead = none ];
229 -> 47;

230 [label="EVAL with clause\nappend2(.(X133, X134), X135, .(X133, X136)) :- append2(X134, X135, X136).\nand substitutionX133 -> T87,\nX134 -> X138,\nX117 -> .(T87, X138),\nT69 -> T85,\nT70 -> .(T87, T88),\nX135 -> .(T85, .(T87, T88)),\nX137 -> T87,\nX136 -> T88,\nT86 -> .(T87, T88)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 230 [arrowhead = none ];
230 -> 48;

231 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 231 [arrowhead = none ];
231 -> 49;

232 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 232 [arrowhead = none ];
232 -> 50;

233 [label="BACKTRACK\nfor clause: append2([], Ys, Ys)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 233 [arrowhead = none ];
233 -> 51;

234 [label="EVAL with clause\nappend2(.(X154, X155), X156, .(X154, X157)) :- append2(X155, X156, X157).\nand substitutionX154 -> T111,\nX155 -> X159,\nX138 -> .(T111, X159),\nT85 -> T108,\nT87 -> T109,\nT88 -> .(T111, T112),\nX156 -> .(T108, .(T109, .(T111, T112))),\nX158 -> T111,\nX157 -> T112,\nT110 -> .(T111, T112)", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 234 [arrowhead = none ];
234 -> 52;

235 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 235 [arrowhead = none ];
235 -> 53;

236 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 236 [arrowhead = none ];
236 -> 57;

237 [label="BACKTRACK\nfor clause: append2([], Ys, Ys)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 237 [arrowhead = none ];
237 -> 58;

238 [label="EVAL with clause\nappend2(.(X175, X176), X177, .(X175, X178)) :- append2(X176, X177, X178).\nand substitutionX175 -> T141,\nX176 -> X180,\nX159 -> .(T141, X180),\nT108 -> T137,\nT109 -> T138,\nT111 -> T139,\nT112 -> .(T141, T142),\nX177 -> .(T137, .(T138, .(T139, .(T141, T142)))),\nX179 -> T141,\nX178 -> T142,\nT140 -> .(T141, T142)", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 238 [arrowhead = none ];
238 -> 59;

239 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 239 [arrowhead = none ];
239 -> 60;

240 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 240 [arrowhead = none ];
240 -> 61;

241 [label="BACKTRACK\nfor clause: append2([], Ys, Ys)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 241 [arrowhead = none ];
241 -> 62;

242 [label="EVAL with clause\nappend2(.(X196, X197), X198, .(X196, X199)) :- append2(X197, X198, X199).\nand substitutionX196 -> T177,\nX197 -> X201,\nX180 -> .(T177, X201),\nT137 -> T172,\nT138 -> T173,\nT139 -> T174,\nT141 -> T175,\nT142 -> .(T177, T178),\nX198 -> .(T172, .(T173, .(T174, .(T175, .(T177, T178))))),\nX200 -> T177,\nX199 -> T178,\nT176 -> .(T177, T178)", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 242 [arrowhead = none ];
242 -> 66;

243 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 243 [arrowhead = none ];
243 -> 67;

244 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 244 [arrowhead = none ];
244 -> 68;

245 [label="BACKTRACK\nfor clause: append2([], Ys, Ys)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 245 [arrowhead = none ];
245 -> 69;

246 [label="EVAL with clause\nappend2(.(X217, X218), X219, .(X217, X220)) :- append2(X218, X219, X220).\nand substitutionX217 -> T219,\nX218 -> X222,\nX201 -> .(T219, X222),\nT172 -> T213,\nT173 -> T214,\nT174 -> T215,\nT175 -> T216,\nT177 -> T217,\nT178 -> .(T219, T220),\nX219 -> .(T213, .(T214, .(T215, .(T216, .(T217, .(T219, T220)))))),\nX221 -> T219,\nX220 -> T220,\nT218 -> .(T219, T220)", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 246 [arrowhead = none ];
246 -> 84;

247 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 247 [arrowhead = none ];
247 -> 85;

248 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
84 -> 248 [arrowhead = none ];
248 -> 86;

249 [label="BACKTRACK\nfor clause: append2([], Ys, Ys)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
86 -> 249 [arrowhead = none ];
249 -> 87;

250 [label="EVAL with clause\nappend2(.(X238, X239), X240, .(X238, X241)) :- append2(X239, X240, X241).\nand substitutionX238 -> T267,\nX239 -> X243,\nX222 -> .(T267, X243),\nT213 -> T260,\nT214 -> T261,\nT215 -> T262,\nT216 -> T263,\nT217 -> T264,\nT219 -> T265,\nT220 -> .(T267, T268),\nX240 -> .(T260, .(T261, .(T262, .(T263, .(T264, .(T265, .(T267, T268))))))),\nX242 -> T267,\nX241 -> T268,\nT266 -> .(T267, T268)", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 250 [arrowhead = none ];
250 -> 88;

251 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 251 [arrowhead = none ];
251 -> 89;

252 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
88 -> 252 [arrowhead = none ];
252 -> 106;

253 [label="BACKTRACK\nfor clause: append2([], Ys, Ys)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
106 -> 253 [arrowhead = none ];
253 -> 107;

254 [label="EVAL with clause\nappend2(.(X259, X260), X261, .(X259, X262)) :- append2(X260, X261, X262).\nand substitutionX259 -> T321,\nX260 -> X264,\nX243 -> .(T321, X264),\nT260 -> T313,\nT261 -> T314,\nT262 -> T315,\nT263 -> T316,\nT264 -> T317,\nT265 -> T318,\nT267 -> T319,\nT268 -> .(T321, T322),\nX261 -> .(T313, .(T314, .(T315, .(T316, .(T317, .(T318, .(T319, .(T321, T322)))))))),\nX263 -> T321,\nX262 -> T322,\nT320 -> .(T321, T322)", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 254 [arrowhead = none ];
254 -> 138;

255 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
107 -> 255 [arrowhead = none ];
255 -> 139;

256 [label="GENERALIZATION\nT341 <-- .(T314, .(T315, .(T316, .(T317, .(T318, .(T319, .(T321, T322)))))))\n\nNew Knowledge:\nT341 is ground", fontsize=14, style = filled, fillcolor = violet];
138 -> 256 [arrowhead = none ];
256 -> 140;

257 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
140 -> 257 [arrowhead = none ];
257 -> 141;

258 [label="EVAL with clause\nappend2([], X271, X271).\nand substitutionX264 -> [],\nT313 -> T350,\nT341 -> T351,\nX271 -> .(T350, T351),\nT322 -> .(T350, T351)", fontsize=14, style = filled, fillcolor = lightblue];
141 -> 258 [arrowhead = none ];
258 -> 142;

259 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
141 -> 259 [arrowhead = none ];
259 -> 143;

260 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
142 -> 260 [arrowhead = none ];
260 -> 144;

261 [label="EVAL with clause\nappend2(.(X306, X307), X308, .(X306, X309)) :- append2(X307, X308, X309).\nand substitutionX306 -> T376,\nX307 -> X311,\nX264 -> .(T376, X311),\nT313 -> T374,\nT341 -> T375,\nX308 -> .(T374, T375),\nX310 -> T376,\nX309 -> T377,\nT322 -> .(T376, T377)", fontsize=14, style = filled, fillcolor = lightblue];
143 -> 261 [arrowhead = none ];
261 -> 146;

262 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
143 -> 262 [arrowhead = none ];
262 -> 147;

263 [label="ONLY EVAL with clause\nappend2(.(X286, X287), X288, .(X286, X289)) :- append2(X287, X288, X289).\nand substitutionX286 -> T360,\nX287 -> X291,\nX264 -> .(T360, X291),\nT350 -> T360,\nT351 -> T361,\nX288 -> .(T360, T361),\nX290 -> T360,\nX289 -> T361", fontsize=14, style = filled, fillcolor = lightblue];
144 -> 263 [arrowhead = none ];
263 -> 145;

264 [label="INSTANCE with matching:\nX117 -> X291\nT69 -> T360\nT70 -> T361", fontsize=14, style = filled, fillcolor = lightgrey];
145 -> 264 [arrowhead = none , style=dashed];
264 -> 45[style=dashed];

265 [label="INSTANCE with matching:\nX264 -> X311\nT313 -> T374\nT341 -> T375\nT322 -> T377", fontsize=14, style = filled, fillcolor = lightgrey];
146 -> 265 [arrowhead = none , style=dashed];
265 -> 140[style=dashed];

266 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
167 -> 266 [arrowhead = none ];
266 -> 168;

267 [label="EVAL with clause\nappend2([], X338, X338).\nand substitutionX333 -> [],\nT390 -> T397,\nX338 -> T397,\nT392 -> T397", fontsize=14, style = filled, fillcolor = lightblue];
168 -> 267 [arrowhead = none ];
267 -> 169;

268 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
168 -> 268 [arrowhead = none ];
268 -> 170;

269 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
169 -> 269 [arrowhead = none ];
269 -> 171;

270 [label="EVAL with clause\nappend2(.(X371, X372), X373, .(X371, X374)) :- append2(X372, X373, X374).\nand substitutionX371 -> T418,\nX372 -> X376,\nX333 -> .(T418, X376),\nT390 -> T417,\nX373 -> T417,\nX375 -> T418,\nX374 -> T419,\nT392 -> .(T418, T419)", fontsize=14, style = filled, fillcolor = lightblue];
170 -> 270 [arrowhead = none ];
270 -> 187;

271 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
170 -> 271 [arrowhead = none ];
271 -> 188;

272 [label="EVAL with clause\nappend2(.(X351, X352), X353, .(X351, X354)) :- append2(X352, X353, X354).\nand substitutionX351 -> T405,\nX352 -> X356,\nX333 -> .(T405, X356),\nT397 -> .(T405, T406),\nX353 -> .(T405, T406),\nX355 -> T405,\nX354 -> T406,\nT404 -> .(T405, T406)", fontsize=14, style = filled, fillcolor = lightblue];
171 -> 272 [arrowhead = none ];
272 -> 185;

273 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
171 -> 273 [arrowhead = none ];
273 -> 186;

274 [label="INSTANCE with matching:\nX117 -> X356\nT69 -> T405\nT70 -> T406", fontsize=14, style = filled, fillcolor = lightgrey];
185 -> 274 [arrowhead = none , style=dashed];
274 -> 45[style=dashed];

275 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
187 -> 275 [arrowhead = none ];
275 -> 189;

276 [label="EVAL with clause\nappend2([], X381, X381).\nand substitutionX376 -> [],\nT417 -> T424,\nX381 -> T424,\nT419 -> T424", fontsize=14, style = filled, fillcolor = lightblue];
189 -> 276 [arrowhead = none ];
276 -> 190;

277 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
189 -> 277 [arrowhead = none ];
277 -> 191;

278 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
190 -> 278 [arrowhead = none ];
278 -> 192;

279 [label="EVAL with clause\nappend2(.(X414, X415), X416, .(X414, X417)) :- append2(X415, X416, X417).\nand substitutionX414 -> T445,\nX415 -> X419,\nX376 -> .(T445, X419),\nT417 -> T444,\nX416 -> T444,\nX418 -> T445,\nX417 -> T446,\nT419 -> .(T445, T446)", fontsize=14, style = filled, fillcolor = lightblue];
191 -> 279 [arrowhead = none ];
279 -> 197;

280 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
191 -> 280 [arrowhead = none ];
280 -> 199;

281 [label="EVAL with clause\nappend2(.(X394, X395), X396, .(X394, X397)) :- append2(X395, X396, X397).\nand substitutionX394 -> T432,\nX395 -> X399,\nX376 -> .(T432, X399),\nT424 -> .(T432, T433),\nX396 -> .(T432, T433),\nX398 -> T432,\nX397 -> T433,\nT431 -> .(T432, T433)", fontsize=14, style = filled, fillcolor = lightblue];
192 -> 281 [arrowhead = none ];
281 -> 193;

282 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
192 -> 282 [arrowhead = none ];
282 -> 194;

283 [label="INSTANCE with matching:\nX117 -> X399\nT69 -> T432\nT70 -> T433", fontsize=14, style = filled, fillcolor = lightgrey];
193 -> 283 [arrowhead = none , style=dashed];
283 -> 45[style=dashed];

284 [label="INSTANCE with matching:\nX333 -> X419\nT390 -> T444\nT392 -> T446\nX9 -> X376\nT391 -> T445\nX97 -> X381", fontsize=14, style = filled, fillcolor = lightgrey];
197 -> 284 [arrowhead = none , style=dashed];
284 -> 167[style=dashed];

}

(44) Obligation:

Complex Complexity Dependency Tuples Problem
MAX

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F1_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f27_in(z0), z0, z1, z2), F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))

S tuples:

F1_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F1_IN, F10_IN, F24_IN, U3', F8_IN, F27_IN, U7', F45_IN, F167_IN, F140_IN, U16'

Compound Symbols:

c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F1_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f27_in(z0), z0, z1, z2), F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))

S tuples:

F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f27_in(z0), z0, z1, z2), F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F1_IN, F10_IN, F24_IN, U3', F8_IN, F27_IN, U7', F45_IN, F167_IN, F140_IN, U16'

Compound Symbols:

c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F1_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f27_in(z0), z0, z1, z2), F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))

S tuples:

F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F1_IN, F10_IN, F24_IN, U3', F8_IN, F27_IN, U7', F45_IN, F167_IN, F140_IN, U16'

Compound Symbols:

c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18

(45) MaxProof (BOTH BOUNDS(ID, ID) transformation)

Took the maximum complexity of the problems.

(46) Complex Obligation (MAX)

(47) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F1_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f27_in(z0), z0, z1, z2), F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))

S tuples:

F1_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F1_IN, F10_IN, F24_IN, U3', F8_IN, F27_IN, U7', F45_IN, F167_IN, F140_IN, U16'

Compound Symbols:

c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18

(48) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 4 of 18 dangling nodes:

U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))

(49) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F1_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f27_in(z0), z0, z1, z2), F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))

S tuples:

F1_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F1_IN, F10_IN, F24_IN, U3', F8_IN, F27_IN, U7', F45_IN, F167_IN, F140_IN

Compound Symbols:

c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12

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

Split RHS of tuples not part of any SCC

(51) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F1_IN(z0, z1) → c2(U1'(f8_in(z0, z1), z0, z1))
F1_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(U2'(f24_in(z1, z2, z0), .(z0, z1), z2))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(U5'(f9_in(z0), f10_in(z1, z0), z0, z1))
F8_IN(z0, z1) → c2(F9_IN(z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
F167_IN(.(z0, z1), .(z0, z1)) → c2(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)))
F167_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))

S tuples:

F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F1_IN(z0, z1) → c2(U1'(f8_in(z0, z1), z0, z1))
F1_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(U2'(f24_in(z1, z2, z0), .(z0, z1), z2))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(U5'(f9_in(z0), f10_in(z1, z0), z0, z1))
F8_IN(z0, z1) → c2(F9_IN(z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F24_IN, U3', F27_IN, U7', F45_IN, F167_IN, F140_IN, F1_IN, F10_IN, F8_IN

Compound Symbols:

c6, c7, c1, c4, c, c6, c11, c12, c2

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

Removed 13 trailing tuple parts

(53) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
F1_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
F167_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7
F27_IN(.(z0, z1)) → c1(F27_IN(z1))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(z0, .(z1, .(z2, z3))) → c6(F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
F1_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F167_IN(.(z0, z1), .(z0, z1)) → c2
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2

S tuples:

F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
F1_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7
F1_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F24_IN, F1_IN, F10_IN, F8_IN, F167_IN, U3', F27_IN, U7', F45_IN, F140_IN

Compound Symbols:

c6, c2, c7, c1, c4, c, c11, c12, c2

(54) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(z0, z1) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F1_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F8_IN(z0, z1) → c2
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F8_IN(z0, z1) → c2
F8_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F10_IN(.(z0, z1), z2) → c2
F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7
U7'(f27_out1(z0, z1), z2, z3, z4) → c4

Now S is empty

(55) BOUNDS(O(1), O(1))

(56) Obligation:

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F1_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f27_in(z0), z0, z1, z2), F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))

S tuples:

F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f27_in(z0), z0, z1, z2), F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F1_IN, F10_IN, F24_IN, U3', F8_IN, F27_IN, U7', F45_IN, F167_IN, F140_IN, U16'

Compound Symbols:

c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F1_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f27_in(z0), z0, z1, z2), F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))

S tuples:

F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F1_IN, F10_IN, F24_IN, U3', F8_IN, F27_IN, U7', F45_IN, F167_IN, F140_IN, U16'

Compound Symbols:

c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18

(57) MultiplicationProof (BOTH BOUNDS(ID, ID) transformation)

Multiplied the complexity of the problems.

(58) Complex Obligation (MULT)

(59) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F1_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f27_in(z0), z0, z1, z2), F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))

S tuples:

F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f27_in(z0), z0, z1, z2), F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F1_IN, F10_IN, F24_IN, U3', F8_IN, F27_IN, U7', F45_IN, F167_IN, F140_IN, U16'

Compound Symbols:

c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18

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

Split RHS of tuples not part of any SCC

(61) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
F1_IN(z0, z1) → c2(U1'(f8_in(z0, z1), z0, z1))
F1_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(U2'(f24_in(z1, z2, z0), .(z0, z1), z2))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(U5'(f9_in(z0), f10_in(z1, z0), z0, z1))
F8_IN(z0, z1) → c2(F9_IN(z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
F167_IN(.(z0, z1), .(z0, z1)) → c2(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)))
F167_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))

S tuples:

F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F24_IN, U3', F27_IN, U7', F8_IN, F45_IN, F167_IN, F140_IN, U16', F1_IN, F10_IN

Compound Symbols:

c6, c7, c1, c4, c, c6, c11, c12, c15, c16, c18, c2

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

Removed 16 trailing tuple parts

(63) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
F1_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
F167_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7
F27_IN(.(z0, z1)) → c1(F27_IN(z1))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(z0, .(z1, .(z2, z3))) → c6(F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F1_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F167_IN(.(z0, z1), .(z0, z1)) → c2
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2

S tuples:

F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
F27_IN(.(z0, z1)) → c1(F27_IN(z1))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F24_IN, F1_IN, F10_IN, F8_IN, F167_IN, U3', F27_IN, U7', F45_IN, F140_IN, U16'

Compound Symbols:

c6, c15, c2, c7, c1, c4, c6, c, c11, c12, c16, c18, c2

(64) 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.

F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6

We considered the (Usable) Rules:

f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)

And the Tuples:

F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
F1_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
F167_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7
F27_IN(.(z0, z1)) → c1(F27_IN(z1))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(z0, .(z1, .(z2, z3))) → c6(F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F1_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F167_IN(.(z0, z1), .(z0, z1)) → c2
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2

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

POL(.(x₁, x₂)) = [1]   
POL(F10_IN(x₁, x₂)) = [2] + [2]x₁ + x₂   
POL(F140_IN(x₁, x₂, x₃)) = [3]   
POL(F167_IN(x₁, x₂)) = [2] + [2]x₁   
POL(F1_IN(x₁, x₂)) = [2] + [3]x₁ + [3]x₂   
POL(F24_IN(x₁, x₂, x₃)) = [3] + x₂   
POL(F27_IN(x₁)) = 0   
POL(F45_IN(x₁, x₂)) = [3]   
POL(F8_IN(x₁, x₂)) = [2] + [2]x₁ + [2]x₂   
POL(U16'(x₁, x₂, x₃, x₄)) = x₃   
POL(U3'(x₁, x₂, x₃, x₄)) = 0   
POL(U6(x₁, x₂)) = [2]x₂   
POL(U7'(x₁, x₂, x₃, x₄)) = [2]   
POL([]) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16) = 0   
POL(c18) = 0   
POL(c2) = 0   
POL(c2(x₁)) = x₁   
POL(c4) = 0   
POL(c6) = 0   
POL(c6(x₁)) = x₁   
POL(c7) = 0   
POL(f27_in(x₁)) = [2]   
POL(f27_out1(x₁, x₂)) = 0   

(65) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
F1_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
F167_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7
F27_IN(.(z0, z1)) → c1(F27_IN(z1))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(z0, .(z1, .(z2, z3))) → c6(F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F1_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F167_IN(.(z0, z1), .(z0, z1)) → c2
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2

S tuples:

F27_IN(.(z0, z1)) → c1(F27_IN(z1))

K tuples:

F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6

Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F24_IN, F1_IN, F10_IN, F8_IN, F167_IN, U3', F27_IN, U7', F45_IN, F140_IN, U16'

Compound Symbols:

c6, c15, c2, c7, c1, c4, c6, c, c11, c12, c16, c18, c2

(66) 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.

F27_IN(.(z0, z1)) → c1(F27_IN(z1))

We considered the (Usable) Rules:

f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)

And the Tuples:

F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
F1_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
F167_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7
F27_IN(.(z0, z1)) → c1(F27_IN(z1))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(z0, .(z1, .(z2, z3))) → c6(F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F1_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F167_IN(.(z0, z1), .(z0, z1)) → c2
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2

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

POL(.(x₁, x₂)) = [2] + x₂   
POL(F10_IN(x₁, x₂)) = [1] + [2]x₁ + [3]x₂   
POL(F140_IN(x₁, x₂, x₃)) = [1]   
POL(F167_IN(x₁, x₂)) = x₂   
POL(F1_IN(x₁, x₂)) = [3] + [3]x₁ + [3]x₂   
POL(F24_IN(x₁, x₂, x₃)) = [1] + x₁ + [3]x₂   
POL(F27_IN(x₁)) = x₁   
POL(F45_IN(x₁, x₂)) = [1]   
POL(F8_IN(x₁, x₂)) = [2] + [3]x₁ + [3]x₂   
POL(U16'(x₁, x₂, x₃, x₄)) = [3]x₃   
POL(U3'(x₁, x₂, x₃, x₄)) = x₂ + [3]x₃   
POL(U6(x₁, x₂)) = 0   
POL(U7'(x₁, x₂, x₃, x₄)) = x₂   
POL([]) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16) = 0   
POL(c18) = 0   
POL(c2) = 0   
POL(c2(x₁)) = x₁   
POL(c4) = 0   
POL(c6) = 0   
POL(c6(x₁)) = x₁   
POL(c7) = 0   
POL(f27_in(x₁)) = 0   
POL(f27_out1(x₁, x₂)) = 0   

(67) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
F1_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
F167_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7
F27_IN(.(z0, z1)) → c1(F27_IN(z1))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(z0, .(z1, .(z2, z3))) → c6(F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F1_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F167_IN(.(z0, z1), .(z0, z1)) → c2
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2

S tuples:none
K tuples:

F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F27_IN(.(z0, z1)) → c1(F27_IN(z1))

Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F24_IN, F1_IN, F10_IN, F8_IN, F167_IN, U3', F27_IN, U7', F45_IN, F140_IN, U16'

Compound Symbols:

c6, c15, c2, c7, c1, c4, c6, c, c11, c12, c16, c18, c2

(68) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(69) BOUNDS(O(1), O(1))

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F1_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f27_in(z0), z0, z1, z2), F27_IN(z0))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))

S tuples:

F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F1_IN, F10_IN, F24_IN, U3', F8_IN, F27_IN, U7', F45_IN, F167_IN, F140_IN, U16'

Compound Symbols:

c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18

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

Split RHS of tuples not part of any SCC

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7(U4'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F27_IN(.(z0, z1)) → c1(U6'(f27_in(z1), .(z0, z1)), F27_IN(z1))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4(U8'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
F1_IN(z0, z1) → c2(U1'(f8_in(z0, z1), z0, z1))
F1_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(U2'(f24_in(z1, z2, z0), .(z0, z1), z2))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(U5'(f9_in(z0), f10_in(z1, z0), z0, z1))
F8_IN(z0, z1) → c2(F9_IN(z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
F167_IN(.(z0, z1), .(z0, z1)) → c2(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)))
F167_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))

S tuples:

F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f167_in(z0, z3), z0, .(z1, .(z2, z3))), F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16(U17'(f28_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
F167_IN(.(z0, z1), .(z0, z1)) → c2(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)))
F167_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F24_IN, U3', F27_IN, U7', F8_IN, F45_IN, F167_IN, F140_IN, U16', F1_IN, F10_IN

Compound Symbols:

c6, c7, c1, c4, c, c6, c11, c12, c15, c16, c18, c2

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

Removed 16 trailing tuple parts

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f1_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f1_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f27_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f27_in(z0), z0, z1, z2)
U3(f27_out1(z0, z1), z2, z3, z4) → U4(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f27_in(z0) → f27_out1([], z0)
f27_in(.(z0, z1)) → U6(f27_in(z1), .(z0, z1))
U6(f27_out1(z0, z1), .(z2, z3)) → f27_out1(.(z2, z0), z1)
U7(f27_out1(z0, z1), z2, z3, z4) → U8(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f167_in(z0, z0) → f167_out1([])
f167_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f167_in(z0, .(z1, z0)) → f167_out1(.(z1, []))
f167_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f167_in(z0, .(z1, .(z2, z3))) → U13(f167_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f167_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f167_out1(.(z3, .(z1, z0)))
U13(f167_out1(z0), z1, .(z2, .(z3, z4))) → f167_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f27_out1(z0, z1), z2, z3, z4) → U17(f28_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f28_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)

Tuples:

F24_IN(z0, z1, z2) → c6(U3'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
F1_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f27_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F27_IN(z0))
F167_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f27_out1(z0, z1), z2, z3, z4) → c7
F27_IN(.(z0, z1)) → c1(F27_IN(z1))
U7'(f27_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(z0, .(z1, .(z2, z3))) → c6(F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F1_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F167_IN(.(z0, z1), .(z0, z1)) → c2
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2

S tuples:

F24_IN(z0, z1, z2) → c15(U16'(f27_in(z0), z0, z1, z2))
F167_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F167_IN(z0, .(z1, .(z2, z3))) → c6(F167_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f27_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F167_IN(.(z0, z1), .(z0, z1)) → c2
F167_IN(.(z0, z1), .(z2, .(z0, z1))) → c2

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f27_in, U6, U7, U8, U9, f45_in, U10, f167_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18

Defined Pair Symbols:

F24_IN, F1_IN, F10_IN, F8_IN, F167_IN, U3', F27_IN, U7', F45_IN, F140_IN, U16'

Compound Symbols:

c6, c15, c2, c7, c1, c4, c6, c, c11, c12, c16, c18, c2