Runtime Complexity proof of /tmp/tmpjvKFiC/rotate.pl

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

0 Prolog

↳1 LPReorderTransformerProof (⇔, 0 ms)

↳2 Prolog

↳3 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 75 ms)

↳4 ComplexCdtProblem

    ↳5 MaxProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳6 MAX

        ↳7 CdtProblem

            ↳8 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 4 ms)

            ↳9 CdtProblem

            ↳10 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

            ↳11 CdtProblem

            ↳12 CdtKnowledgeProof (⇔, 0 ms)

            ↳13 BOUNDS(O(1), O(1))

        ↳14 ComplexCdtProblem

            ↳15 MultiplicationProof (BOTH BOUNDS(ID, ID), 0 ms)

            ↳16 MULT

                ↳17 CdtProblem

                    ↳18 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

                    ↳19 CdtProblem

                    ↳20 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

                    ↳21 CdtProblem

                    ↳22 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 328 ms)

                    ↳23 CdtProblem

                    ↳24 CdtKnowledgeProof (⇔, 0 ms)

                    ↳25 CdtProblem

                    ↳26 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 92 ms)

                    ↳27 CdtProblem

                    ↳28 SIsEmptyProof (⇔, 0 ms)

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

                ↳30 CdtProblem

                    ↳31 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

                    ↳32 CdtProblem

                    ↳33 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

                    ↳34 CdtProblem

                    ↳35 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 52 ms)

                    ↳36 CdtProblem

                    ↳37 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 49 ms)

                    ↳38 CdtProblem

                    ↳39 SIsEmptyProof (⇔, 0 ms)

                    ↳40 BOUNDS(O(1), O(1))

↳41 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 158 ms)

↳42 ComplexCdtProblem

    ↳43 MaxProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳44 MAX

        ↳45 CdtProblem

            ↳46 CdtGraphRemoveDanglingProof (BOTH BOUNDS(ID, ID), 0 ms)

            ↳47 CdtProblem

            ↳48 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

            ↳49 CdtProblem

            ↳50 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

            ↳51 CdtProblem

            ↳52 CdtKnowledgeProof (⇔, 0 ms)

            ↳53 CdtProblem

            ↳54 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 294 ms)

            ↳55 CdtProblem

            ↳56 SIsEmptyProof (⇔, 0 ms)

            ↳57 BOUNDS(O(1), O(1))

        ↳58 ComplexCdtProblem

            ↳59 MultiplicationProof (BOTH BOUNDS(ID, ID), 0 ms)

            ↳60 MULT

                ↳61 CdtProblem

                    ↳62 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

                    ↳63 CdtProblem

                    ↳64 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

                    ↳65 CdtProblem

                    ↳66 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 69 ms)

                    ↳67 CdtProblem

                    ↳68 CdtKnowledgeProof (⇔, 0 ms)

                    ↳69 CdtProblem

                    ↳70 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 104 ms)

                    ↳71 CdtProblem

                    ↳72 SIsEmptyProof (⇔, 0 ms)

                    ↳73 BOUNDS(O(1), O(1))

                ↳74 CdtProblem

                    ↳75 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

                    ↳76 CdtProblem

                    ↳77 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

                    ↳78 CdtProblem

(0) Obligation:

Clauses:

rotate(X, Y) :- ','(append(A, B, X), append(B, A, Y)).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).
append([], Ys, Ys).

Query: rotate(g,a)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

append([], Ys, Ys).
rotate(X, Y) :- ','(append(A, B, X), append(B, A, Y)).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).

Query: rotate(g,a)

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

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: rotate(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: rotate(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(append(X13, X14, T10), append(X14, X13, T12))\n\nT10 is ground\nX13 is free\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: append(X13, X14, T10)\n\nT10 is ground\nX13 is free\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: append(T16, T15, T12)\n\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: append(X13, X14, T10), SCOPE: 2, CLAUSE: 0 | append(X13, X14, T10), SCOPE: 2, CLAUSE: 2\n\nT10 is ground\nX13 is free\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true | append(X13, X14, T20), SCOPE: 2, CLAUSE: 2\n\nT20 is ground\nX13 is free\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: append(X13, X14, T20), SCOPE: 2, CLAUSE: 2\n\nT20 is ground\nX13 is free\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: append(X46, X47, T26)\n\nT26 is ground\nX46 is free\nX47 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: append(T16, T15, T12), SCOPE: 3, CLAUSE: 0 | append(T16, T15, T12), SCOPE: 3, CLAUSE: 2\n\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true | append([], T32, T12), SCOPE: 3, CLAUSE: 2\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: append(T16, T15, T12), SCOPE: 3, CLAUSE: 2\n\nT15 is ground\nT16 is ground\nX54 is free\nappend(T16, T15, T12) !=? append([], X54, X54)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: append([], T32, T12), SCOPE: 3, CLAUSE: 2\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: append(T42, T43, T45)\n\nT42 is ground\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 36 [arrowhead = none ];
36 -> 4;

37 [label="ONLY EVAL with clause\nrotate(X11, X12) :- ','(append(X13, X14, X11), append(X14, X13, X12)).\nand substitutionT1 -> T10,\nX11 -> T10,\nT2 -> T12,\nX12 -> T12,\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 37 [arrowhead = none ];
37 -> 7;

38 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 38 [arrowhead = none ];
38 -> 12;

39 [label="SPLIT 2\nnew knowledge:\nT15 is ground\nT16 is ground\nT10 is ground\nreplacements:X13 -> T15,\nX14 -> T16", fontsize=14, style = filled, fillcolor = violet];
7 -> 39 [arrowhead = none ];
39 -> 13;

40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 40 [arrowhead = none ];
40 -> 17;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 41 [arrowhead = none ];
41 -> 25;

42 [label="ONLY EVAL with clause\nappend([], X25, X25).\nand substitutionX13 -> [],\nX14 -> T20,\nX25 -> T20,\nT10 -> T20,\nX26 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 42 [arrowhead = none ];
42 -> 19;

43 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 43 [arrowhead = none ];
43 -> 20;

44 [label="EVAL with clause\nappend(.(X41, X42), X43, .(X41, X44)) :- append(X42, X43, X44).\nand substitutionX41 -> T25,\nX42 -> X46,\nX13 -> .(T25, X46),\nX14 -> X47,\nX43 -> X47,\nX45 -> T25,\nX44 -> T26,\nT20 -> .(T25, T26)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 44 [arrowhead = none ];
44 -> 23;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 45 [arrowhead = none ];
45 -> 24;

46 [label="INSTANCE with matching:\nX13 -> X46\nX14 -> X47\nT10 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 46 [arrowhead = none , style=dashed];
46 -> 12[style=dashed];

47 [label="EVAL with clause\nappend([], X54, X54).\nand substitutionT16 -> [],\nT15 -> T32,\nX54 -> T32,\nT12 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 47 [arrowhead = none ];
47 -> 26;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 48 [arrowhead = none ];
48 -> 27;

49 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 49 [arrowhead = none ];
49 -> 28;

50 [label="EVAL with clause\nappend(.(X67, X68), X69, .(X67, X70)) :- append(X68, X69, X70).\nand substitutionX67 -> T41,\nX68 -> T42,\nT16 -> .(T41, T42),\nT15 -> T43,\nX69 -> T43,\nX70 -> T45,\nT12 -> .(T41, T45),\nT44 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 50 [arrowhead = none ];
50 -> 33;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 51 [arrowhead = none ];
51 -> 35;

52 [label="BACKTRACK\nfor clause: append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 52 [arrowhead = none ];
52 -> 29;

53 [label="INSTANCE with matching:\nT16 -> T42\nT15 -> T43\nT12 -> T45", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 53 [arrowhead = none , style=dashed];
53 -> 13[style=dashed];

}

(4) Obligation:

Complex Complexity Dependency Tuples Problem
MAX

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
Tuples:

F1_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0) → c3(U5'(f12_in(z0), z0), F12_IN(z0))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c4(U9'(f13_in(z1, z0), z2, z0, z1), F13_IN(z1, z0))
S tuples:

F1_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))
K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F1_IN, F7_IN, U2', F12_IN, U5', F13_IN, U8'

Compound Symbols:

c, c2, c3, c1, c3, c4, c4

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
Tuples:

F1_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0) → c3(U5'(f12_in(z0), z0), F12_IN(z0))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c4(U9'(f13_in(z1, z0), z2, z0, z1), F13_IN(z1, z0))
S tuples:

F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0) → c3(U5'(f12_in(z0), z0), F12_IN(z0))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F1_IN, F7_IN, U2', F12_IN, U5', F13_IN, U8'

Compound Symbols:

c, c2, c3, c1, c3, c4, c4

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
Tuples:

F1_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0) → c3(U5'(f12_in(z0), z0), F12_IN(z0))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c4(U9'(f13_in(z1, z0), z2, z0, z1), F13_IN(z1, z0))
S tuples:

F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c4(U9'(f13_in(z1, z0), z2, z0, z1), F13_IN(z1, z0))
K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F1_IN, F7_IN, U2', F12_IN, U5', F13_IN, U8'

Compound Symbols:

c, c2, c3, c1, c3, c4, c4

(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:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F1_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0) → c3(U5'(f12_in(z0), z0), F12_IN(z0))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c4(U9'(f13_in(z1, z0), z2, z0, z1), F13_IN(z1, z0))

S tuples:

F1_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F1_IN, F7_IN, U2', F12_IN, U5', F13_IN, U8'

Compound Symbols:

c, c2, c3, c1, c3, c4, c4

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

Split RHS of tuples not part of any SCC

(9) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(U1'(f7_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(U9'(f13_in(z1, z0), z2, z0, z1))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))

S tuples:

F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))
F1_IN(z0) → c5(U1'(f7_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F7_IN, U2', F12_IN, U5', F13_IN, F1_IN, U8'

Compound Symbols:

c2, c3, c1, c4, c5

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

Removed 6 trailing tuple parts

(11) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F7_IN(z0) → c2(U2'(f12_in(z0), z0))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U2'(f12_out1(z0, z1), z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))
F1_IN(z0) → c5
U8'(f12_out1(z0, z1), z2) → c5

S tuples:

F7_IN(z0) → c2(U2'(f12_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
U2'(f12_out1(z0, z1), z2) → c3
F1_IN(z0) → c5

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F7_IN, F1_IN, U8', U2', F12_IN, U5', F13_IN

Compound Symbols:

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

(12) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(z0) → c5(F7_IN(z0))
F1_IN(z0) → c5
F7_IN(z0) → c2(U2'(f12_in(z0), z0))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U2'(f12_out1(z0, z1), z2) → c3
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U8'(f12_out1(z0, z1), z2) → c5
U5'(f12_out1(z0, z1), z2) → c4

Now S is empty

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

(14) Obligation:

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
Tuples:

F1_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0) → c3(U5'(f12_in(z0), z0), F12_IN(z0))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c4(U9'(f13_in(z1, z0), z2, z0, z1), F13_IN(z1, z0))
S tuples:

F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0) → c3(U5'(f12_in(z0), z0), F12_IN(z0))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F1_IN, F7_IN, U2', F12_IN, U5', F13_IN, U8'

Compound Symbols:

c, c2, c3, c1, c3, c4, c4

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
Tuples:

F1_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0) → c3(U5'(f12_in(z0), z0), F12_IN(z0))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c4(U9'(f13_in(z1, z0), z2, z0, z1), F13_IN(z1, z0))
S tuples:

F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c4(U9'(f13_in(z1, z0), z2, z0, z1), F13_IN(z1, z0))
K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F1_IN, F7_IN, U2', F12_IN, U5', F13_IN, U8'

Compound Symbols:

c, c2, c3, c1, c3, c4, c4

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

Multiplied the complexity of the problems.

(16) Complex Obligation (MULT)

(17) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F1_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0) → c3(U5'(f12_in(z0), z0), F12_IN(z0))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c4(U9'(f13_in(z1, z0), z2, z0, z1), F13_IN(z1, z0))

S tuples:

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

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F1_IN, F7_IN, U2', F12_IN, U5', F13_IN, U8'

Compound Symbols:

c, c2, c3, c1, c3, c4, c4

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

Split RHS of tuples not part of any SCC

(19) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(U1'(f7_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(U9'(f13_in(z1, z0), z2, z0, z1))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))

S tuples:

F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F7_IN, U2', F12_IN, U5', F13_IN, F1_IN, U8'

Compound Symbols:

c2, c3, c1, c4, c5

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

Removed 6 trailing tuple parts

(21) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F7_IN(z0) → c2(U2'(f12_in(z0), z0))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U2'(f12_out1(z0, z1), z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))
F1_IN(z0) → c5
U8'(f12_out1(z0, z1), z2) → c5

S tuples:

F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F7_IN, F1_IN, U8', U2', F12_IN, U5', F13_IN

Compound Symbols:

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

(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.

F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(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) → c2(U2'(f12_in(z0), z0))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U2'(f12_out1(z0, z1), z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))
F1_IN(z0) → c5
U8'(f12_out1(z0, z1), z2) → c5

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

POL(.(x₁, x₂)) = x₂
POL(F12_IN(x₁)) = [3]x₁
POL(F13_IN(x₁, x₂)) = [1]
POL(F1_IN(x₁)) = [2] + [3]x₁
POL(F7_IN(x₁)) = [1] + [3]x₁
POL(U2'(x₁, x₂)) = x₂
POL(U4(x₁, x₂)) = 0
POL(U5'(x₁, x₂)) = 0
POL(U8'(x₁, x₂)) = [1] + [3]x₂
POL([]) = 0
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3) = 0
POL(c3(x₁)) = x₁
POL(c4) = 0
POL(c5) = 0
POL(c5(x₁)) = x₁
POL(f12_in(x₁)) = 0
POL(f12_out1(x₁, x₂)) = 0

(23) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F7_IN(z0) → c2(U2'(f12_in(z0), z0))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U2'(f12_out1(z0, z1), z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))
F1_IN(z0) → c5
U8'(f12_out1(z0, z1), z2) → c5

S tuples:

F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4

K tuples:

F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))

Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F7_IN, F1_IN, U8', U2', F12_IN, U5', F13_IN

Compound Symbols:

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

(24) CdtKnowledgeProof (EQUIVALENT transformation)

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

U5'(f12_out1(z0, z1), z2) → c4

(25) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F7_IN(z0) → c2(U2'(f12_in(z0), z0))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U2'(f12_out1(z0, z1), z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))
F1_IN(z0) → c5
U8'(f12_out1(z0, z1), z2) → c5

S tuples:

F12_IN(.(z0, z1)) → c1(F12_IN(z1))

K tuples:

F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U5'(f12_out1(z0, z1), z2) → c4

Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F7_IN, F1_IN, U8', U2', F12_IN, U5', F13_IN

Compound Symbols:

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

(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) → c2(U2'(f12_in(z0), z0))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U2'(f12_out1(z0, z1), z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))
F1_IN(z0) → c5
U8'(f12_out1(z0, z1), z2) → c5

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

POL(.(x₁, x₂)) = [2] + x₂
POL(F12_IN(x₁)) = [2]x₁
POL(F13_IN(x₁, x₂)) = 0
POL(F1_IN(x₁)) = [2] + [2]x₁
POL(F7_IN(x₁)) = [2]x₁
POL(U2'(x₁, x₂)) = 0
POL(U4(x₁, x₂)) = 0
POL(U5'(x₁, x₂)) = x₂
POL(U8'(x₁, x₂)) = x₂
POL([]) = 0
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3) = 0
POL(c3(x₁)) = x₁
POL(c4) = 0
POL(c5) = 0
POL(c5(x₁)) = x₁
POL(f12_in(x₁)) = 0
POL(f12_out1(x₁, x₂)) = 0

(27) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F7_IN(z0) → c2(U2'(f12_in(z0), z0))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U2'(f12_out1(z0, z1), z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))
F1_IN(z0) → c5
U8'(f12_out1(z0, z1), z2) → c5

S tuples:none
K tuples:

F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U5'(f12_out1(z0, z1), z2) → c4
F12_IN(.(z0, z1)) → c1(F12_IN(z1))

Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F7_IN, F1_IN, U8', U2', F12_IN, U5', F13_IN

Compound Symbols:

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

(28) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F1_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0) → c3(U5'(f12_in(z0), z0), F12_IN(z0))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c4(U9'(f13_in(z1, z0), z2, z0, z1), F13_IN(z1, z0))

S tuples:

F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c4(U9'(f13_in(z1, z0), z2, z0, z1), F13_IN(z1, z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F1_IN, F7_IN, U2', F12_IN, U5', F13_IN, U8'

Compound Symbols:

c, c2, c3, c1, c3, c4, c4

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

Split RHS of tuples not part of any SCC

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F7_IN(z0) → c2(U2'(f12_in(z0), z0))
U2'(f12_out1(z0, z1), z2) → c3(U3'(f13_in(z1, z0), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4(U6'(f13_in(z1, z0), z2, z0, z1))
F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(U1'(f7_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(U9'(f13_in(z1, z0), z2, z0, z1))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))

S tuples:

F13_IN(.(z0, z1), z2) → c1(U7'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c5(U9'(f13_in(z1, z0), z2, z0, z1))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F7_IN, U2', F12_IN, U5', F13_IN, F1_IN, U8'

Compound Symbols:

c2, c3, c1, c4, c5

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

Removed 6 trailing tuple parts

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F7_IN(z0) → c2(U2'(f12_in(z0), z0))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U2'(f12_out1(z0, z1), z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))
F1_IN(z0) → c5
U8'(f12_out1(z0, z1), z2) → c5

S tuples:

F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))
U8'(f12_out1(z0, z1), z2) → c5

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F7_IN, F1_IN, U8', U2', F12_IN, U5', F13_IN

Compound Symbols:

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

(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) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U8'(f12_out1(z0, z1), z2) → c5

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) → c2(U2'(f12_in(z0), z0))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U2'(f12_out1(z0, z1), z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))
F1_IN(z0) → c5
U8'(f12_out1(z0, z1), z2) → c5

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

POL(.(x₁, x₂)) = x₂
POL(F12_IN(x₁)) = [1] + [2]x₁
POL(F13_IN(x₁, x₂)) = 0
POL(F1_IN(x₁)) = [2] + [3]x₁
POL(F7_IN(x₁)) = [2] + [2]x₁
POL(U2'(x₁, x₂)) = [2]x₂
POL(U4(x₁, x₂)) = 0
POL(U5'(x₁, x₂)) = [2]x₂
POL(U8'(x₁, x₂)) = [1] + x₂
POL([]) = 0
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3) = 0
POL(c3(x₁)) = x₁
POL(c4) = 0
POL(c5) = 0
POL(c5(x₁)) = x₁
POL(f12_in(x₁)) = 0
POL(f12_out1(x₁, x₂)) = 0

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F7_IN(z0) → c2(U2'(f12_in(z0), z0))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U2'(f12_out1(z0, z1), z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))
F1_IN(z0) → c5
U8'(f12_out1(z0, z1), z2) → c5

S tuples:

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

K tuples:

F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U8'(f12_out1(z0, z1), z2) → c5

Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F7_IN, F1_IN, U8', U2', F12_IN, U5', F13_IN

Compound Symbols:

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

(37) 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), z2) → c1(F13_IN(z1, 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) → c2(U2'(f12_in(z0), z0))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U2'(f12_out1(z0, z1), z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))
F1_IN(z0) → c5
U8'(f12_out1(z0, z1), z2) → c5

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
POL(F1_IN(x₁)) = [2] + [3]x₁
POL(F7_IN(x₁)) = [2] + [3]x₁
POL(U2'(x₁, x₂)) = [1] + x₁ + [2]x₂
POL(U4(x₁, x₂)) = x₁
POL(U5'(x₁, x₂)) = [1]
POL(U8'(x₁, x₂)) = [1] + [2]x₁
POL([]) = 0
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3) = 0
POL(c3(x₁)) = x₁
POL(c4) = 0
POL(c5) = 0
POL(c5(x₁)) = x₁
POL(f12_in(x₁)) = x₁
POL(f12_out1(x₁, x₂)) = x₂

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0, z1, z2), z3) → f1_out1(z2)
f7_in(z0) → U2(f12_in(z0), z0)
f7_in(z0) → U5(f12_in(z0), z0)
f7_in(z0) → U8(f12_in(z0), z0)
U2(f12_out1(z0, z1), z2) → U3(f13_in(z1, z0), z2, z0, z1)
U3(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, 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) → U6(f13_in(z1, z0), z2, z0, z1)
U6(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U7(f13_in(z1, z2), .(z0, z1), z2)
U7(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
U8(f12_out1(z0, z1), z2) → U9(f13_in(z1, z0), z2, z0, z1)
U9(f13_out1(z0), z1, z2, z3) → f7_out1(z2, z3, z0)

Tuples:

F7_IN(z0) → c2(U2'(f12_in(z0), z0))
F7_IN(z0) → c3(U8'(f12_in(z0), z0))
F1_IN(z0) → c5(F7_IN(z0))
F7_IN(z0) → c5(U5'(f12_in(z0), z0))
F7_IN(z0) → c5(F12_IN(z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U2'(f12_out1(z0, z1), z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0, z1), z2) → c4
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))
F1_IN(z0) → c5
U8'(f12_out1(z0, z1), z2) → c5

S tuples:none
K tuples:

F7_IN(z0) → c3(U8'(f12_in(z0), z0))
U8'(f12_out1(z0, z1), z2) → c5(F13_IN(z1, z0))
U8'(f12_out1(z0, z1), z2) → c5
F13_IN(.(z0, z1), z2) → c1(F13_IN(z1, z2))

Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9

Defined Pair Symbols:

F7_IN, F1_IN, U8', U2', F12_IN, U5', F13_IN

Compound Symbols:

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

(39) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(40) BOUNDS(O(1), O(1))

(41) 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: rotate(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: rotate(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(append(X5, X6, T5), append(X6, X5, T7))\n\nT5 is ground\nX5 is free\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(append(X5, X6, T5), append(X6, X5, T7)), SCOPE: 2, CLAUSE: 0 | ','(append(X5, X6, T5), append(X6, X5, T7)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nX5 is free\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: append(T10, [], T7) | ','(append(X5, X6, T10), append(X6, X5, T7)), SCOPE: 2, CLAUSE: 2\n\nT10 is ground\nX5 is free\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: append(T10, [], T7)\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(append(X5, X6, T10), append(X6, X5, T7)), SCOPE: 2, CLAUSE: 2\n\nT10 is ground\nX5 is free\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: append(T10, [], T7), SCOPE: 3, CLAUSE: 0 | append(T10, [], T7), SCOPE: 3, CLAUSE: 2\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true | append([], [], T7), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: append(T10, [], T7), SCOPE: 3, CLAUSE: 2\n\nT10 is ground\nX21 is free\nappend(T10, [], T7) !=? append([], X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: append([], [], T7), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: append(T18, [], T20)\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(append(X59, X60, T26), append(X60, .(T25, X59), T7))\n\nT25 is ground\nT26 is ground\nX59 is free\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: append(X59, X60, T26)\n\nT26 is ground\nX59 is free\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: append(T33, .(T25, T32), T7)\n\nT25 is ground\nT32 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: append(X59, X60, T26), SCOPE: 4, CLAUSE: 0 | append(X59, X60, T26), SCOPE: 4, CLAUSE: 2\n\nT26 is ground\nX59 is free\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true | append(X59, X60, T37), SCOPE: 4, CLAUSE: 2\n\nT37 is ground\nX59 is free\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: append(X59, X60, T37), SCOPE: 4, CLAUSE: 2\n\nT37 is ground\nX59 is free\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: append(X94, X95, T43)\n\nT43 is ground\nX94 is free\nX95 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: append(T33, .(T25, T32), T7), SCOPE: 5, CLAUSE: 0 | append(T33, .(T25, T32), T7), SCOPE: 5, CLAUSE: 2\n\nT25 is ground\nT32 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true | append([], .(T53, T54), T7), SCOPE: 5, CLAUSE: 2\n\nT53 is ground\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: append(T33, .(T25, T32), T7), SCOPE: 5, CLAUSE: 2\n\nT25 is ground\nT32 is ground\nT33 is ground\nX102 is free\nappend(T33, .(T25, T32), T7) !=? append([], X102, X102)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: append([], .(T53, T54), T7), SCOPE: 5, CLAUSE: 2\n\nT53 is ground\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: append(T66, .(T67, T68), T70)\n\nT66 is ground\nT67 is ground\nT68 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 48 [arrowhead = none ];
48 -> 3;

49 [label="ONLY EVAL with clause\nrotate(X3, X4) :- ','(append(X5, X6, X3), append(X6, X5, X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 49 [arrowhead = none ];
49 -> 5;

50 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 50 [arrowhead = none ];
50 -> 6;

51 [label="ONLY EVAL with clause\nappend([], X13, X13).\nand substitutionX5 -> [],\nX6 -> T10,\nX13 -> T10,\nT5 -> T10,\nX14 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 51 [arrowhead = none ];
51 -> 8;

52 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 52 [arrowhead = none ];
52 -> 9;

53 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 53 [arrowhead = none ];
53 -> 10;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 54 [arrowhead = none ];
54 -> 11;

55 [label="EVAL with clause\nappend(.(X54, X55), X56, .(X54, X57)) :- append(X55, X56, X57).\nand substitutionX54 -> T25,\nX55 -> X59,\nX5 -> .(T25, X59),\nX6 -> X60,\nX56 -> X60,\nX58 -> T25,\nX57 -> T26,\nT10 -> .(T25, T26)", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 55 [arrowhead = none ];
55 -> 30;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 56 [arrowhead = none ];
56 -> 31;

57 [label="EVAL with clause\nappend([], X21, X21).\nand substitutionT10 -> [],\nX21 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 57 [arrowhead = none ];
57 -> 14;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 58 [arrowhead = none ];
58 -> 15;

59 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 59 [arrowhead = none ];
59 -> 16;

60 [label="EVAL with clause\nappend(.(X34, X35), X36, .(X34, X37)) :- append(X35, X36, X37).\nand substitutionX34 -> T17,\nX35 -> T18,\nT10 -> .(T17, T18),\nX36 -> [],\nX37 -> T20,\nT7 -> .(T17, T20),\nT19 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 60 [arrowhead = none ];
60 -> 21;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 61 [arrowhead = none ];
61 -> 22;

62 [label="BACKTRACK\nfor clause: append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 62 [arrowhead = none ];
62 -> 18;

63 [label="INSTANCE with matching:\nT10 -> T18\nT7 -> T20", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 63 [arrowhead = none , style=dashed];
63 -> 9[style=dashed];

64 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
30 -> 64 [arrowhead = none ];
64 -> 32;

65 [label="SPLIT 2\nnew knowledge:\nT32 is ground\nT33 is ground\nT26 is ground\nreplacements:X59 -> T32,\nX60 -> T33", fontsize=14, style = filled, fillcolor = violet];
30 -> 65 [arrowhead = none ];
65 -> 34;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 66 [arrowhead = none ];
66 -> 36;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 67 [arrowhead = none ];
67 -> 41;

68 [label="ONLY EVAL with clause\nappend([], X73, X73).\nand substitutionX59 -> [],\nX60 -> T37,\nX73 -> T37,\nT26 -> T37,\nX74 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 68 [arrowhead = none ];
68 -> 37;

69 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 69 [arrowhead = none ];
69 -> 38;

70 [label="EVAL with clause\nappend(.(X89, X90), X91, .(X89, X92)) :- append(X90, X91, X92).\nand substitutionX89 -> T42,\nX90 -> X94,\nX59 -> .(T42, X94),\nX60 -> X95,\nX91 -> X95,\nX93 -> T42,\nX92 -> T43,\nT37 -> .(T42, T43)", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 70 [arrowhead = none ];
70 -> 39;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 71 [arrowhead = none ];
71 -> 40;

72 [label="INSTANCE with matching:\nX59 -> X94\nX60 -> X95\nT26 -> T43", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 72 [arrowhead = none , style=dashed];
72 -> 32[style=dashed];

73 [label="EVAL with clause\nappend([], X102, X102).\nand substitutionT33 -> [],\nT25 -> T53,\nT32 -> T54,\nX102 -> .(T53, T54),\nT7 -> .(T53, T54)", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 73 [arrowhead = none ];
73 -> 42;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 74 [arrowhead = none ];
74 -> 43;

75 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 75 [arrowhead = none ];
75 -> 44;

76 [label="EVAL with clause\nappend(.(X115, X116), X117, .(X115, X118)) :- append(X116, X117, X118).\nand substitutionX115 -> T65,\nX116 -> T66,\nT33 -> .(T65, T66),\nT25 -> T67,\nT32 -> T68,\nX117 -> .(T67, T68),\nX118 -> T70,\nT7 -> .(T65, T70),\nT69 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 76 [arrowhead = none ];
76 -> 46;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 77 [arrowhead = none ];
77 -> 47;

78 [label="BACKTRACK\nfor clause: append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 78 [arrowhead = none ];
78 -> 45;

79 [label="INSTANCE with matching:\nT33 -> T66\nT25 -> T67\nT32 -> T68\nT7 -> T70", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 79 [arrowhead = none , style=dashed];
79 -> 34[style=dashed];

}

(42) Obligation:

Complex Complexity Dependency Tuples Problem
MAX

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)
Tuples:

F2_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c6(U3'(f30_in(z1, z0), .(z0, z1)), F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c11(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))
F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F30_IN(z0, z1) → c3(U8'(f32_in(z0), z0, z1), F32_IN(z0))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))
F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c4(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1), F34_IN(z1, z3, z0))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))
S tuples:

F2_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c6(U3'(f30_in(z1, z0), .(z0, z1)), F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c11(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))
K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F2_IN, F9_IN, F10_IN, F30_IN, U4', F8_IN, F32_IN, U8', F34_IN, U12'

Compound Symbols:

c, c4, c6, c8, c9, c11, c1, c3, c4, c6, c3

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)
Tuples:

F2_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c6(U3'(f30_in(z1, z0), .(z0, z1)), F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c11(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))
F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F30_IN(z0, z1) → c3(U8'(f32_in(z0), z0, z1), F32_IN(z0))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))
F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c4(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1), F34_IN(z1, z3, z0))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))
S tuples:

F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F30_IN(z0, z1) → c3(U8'(f32_in(z0), z0, z1), F32_IN(z0))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))
K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F2_IN, F9_IN, F10_IN, F30_IN, U4', F8_IN, F32_IN, U8', F34_IN, U12'

Compound Symbols:

c, c4, c6, c8, c9, c11, c1, c3, c4, c6, c3

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)
Tuples:

F2_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c6(U3'(f30_in(z1, z0), .(z0, z1)), F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c11(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))
F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F30_IN(z0, z1) → c3(U8'(f32_in(z0), z0, z1), F32_IN(z0))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))
F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c4(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1), F34_IN(z1, z3, z0))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))
S tuples:

F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c4(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1), F34_IN(z1, z3, z0))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))
K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F2_IN, F9_IN, F10_IN, F30_IN, U4', F8_IN, F32_IN, U8', F34_IN, U12'

Compound Symbols:

c, c4, c6, c8, c9, c11, c1, c3, c4, c6, c3

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

Took the maximum complexity of the problems.

(44) Complex Obligation (MAX)

(45) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F2_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c6(U3'(f30_in(z1, z0), .(z0, z1)), F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c11(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))
F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F30_IN(z0, z1) → c3(U8'(f32_in(z0), z0, z1), F32_IN(z0))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))
F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c4(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1), F34_IN(z1, z3, z0))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))

S tuples:

F2_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c6(U3'(f30_in(z1, z0), .(z0, z1)), F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c11(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F2_IN, F9_IN, F10_IN, F30_IN, U4', F8_IN, F32_IN, U8', F34_IN, U12'

Compound Symbols:

c, c4, c6, c8, c9, c11, c1, c3, c4, c6, c3

(46) CdtGraphRemoveDanglingProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 of 14 dangling nodes:

F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))

(47) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F2_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c6(U3'(f30_in(z1, z0), .(z0, z1)), F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c11(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))
F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F30_IN(z0, z1) → c3(U8'(f32_in(z0), z0, z1), F32_IN(z0))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c4(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1), F34_IN(z1, z3, z0))

S tuples:

F2_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c6(U3'(f30_in(z1, z0), .(z0, z1)), F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c11(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F2_IN, F9_IN, F10_IN, F30_IN, U4', F8_IN, F32_IN, U8', F34_IN, U12'

Compound Symbols:

c, c4, c6, c8, c9, c11, c1, c3, c4, c3

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

Split RHS of tuples not part of any SCC

(49) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F2_IN(z0) → c2(U1'(f8_in(z0), z0))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(U3'(f30_in(z1, z0), .(z0, z1)))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(U6'(f9_in(z0), f10_in(z0), z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))

S tuples:

F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F2_IN(z0) → c2(U1'(f8_in(z0), z0))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(U3'(f30_in(z1, z0), .(z0, z1)))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(U6'(f9_in(z0), f10_in(z0), z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F9_IN, F30_IN, U4', F32_IN, U8', F34_IN, F2_IN, F10_IN, F8_IN, U12'

Compound Symbols:

c4, c8, c9, c1, c4, c3, c2

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

Removed 9 trailing tuple parts

(51) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
F9_IN(.(z0, z1)) → c4(F9_IN(z1))
U4'(f32_out1(z0, z1), z2, z3) → c9
F32_IN(.(z0, z1)) → c1(F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4
F34_IN(.(z0, z1), z2, z3) → c1(F34_IN(z1, z2, z3))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
U12'(f32_out1(z0, z1), z2, z3) → c2

S tuples:

F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F9_IN(.(z0, z1)) → c4(F9_IN(z1))
U4'(f32_out1(z0, z1), z2, z3) → c9
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2

K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F30_IN, F2_IN, F10_IN, F8_IN, U12', F9_IN, U4', F32_IN, U8', F34_IN

Compound Symbols:

c8, c3, c2, c4, c9, c1, c4, c2

(52) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN(z0) → c2(F8_IN(z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F8_IN(z0) → c2
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F10_IN(.(z0, z1)) → c2
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U4'(f32_out1(z0, z1), z2, z3) → c9
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
U12'(f32_out1(z0, z1), z2, z3) → c2
U8'(f32_out1(z0, z1), z2, z3) → c4

(53) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
F9_IN(.(z0, z1)) → c4(F9_IN(z1))
U4'(f32_out1(z0, z1), z2, z3) → c9
F32_IN(.(z0, z1)) → c1(F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4
F34_IN(.(z0, z1), z2, z3) → c1(F34_IN(z1, z2, z3))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
U12'(f32_out1(z0, z1), z2, z3) → c2

S tuples:

F9_IN(.(z0, z1)) → c4(F9_IN(z1))

K tuples:

F2_IN(z0) → c2(F8_IN(z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U4'(f32_out1(z0, z1), z2, z3) → c9
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
U12'(f32_out1(z0, z1), z2, z3) → c2
U8'(f32_out1(z0, z1), z2, z3) → c4

Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F30_IN, F2_IN, F10_IN, F8_IN, U12', F9_IN, U4', F32_IN, U8', F34_IN

Compound Symbols:

c8, c3, c2, c4, c9, c1, c4, c2

(54) 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.

F9_IN(.(z0, z1)) → c4(F9_IN(z1))

We considered the (Usable) Rules:

f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)

And the Tuples:

F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
F9_IN(.(z0, z1)) → c4(F9_IN(z1))
U4'(f32_out1(z0, z1), z2, z3) → c9
F32_IN(.(z0, z1)) → c1(F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4
F34_IN(.(z0, z1), z2, z3) → c1(F34_IN(z1, z2, z3))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
U12'(f32_out1(z0, z1), z2, z3) → c2

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

POL(.(x₁, x₂)) = [2] + x₂
POL(F10_IN(x₁)) = x₁
POL(F2_IN(x₁)) = [1] + [3]x₁
POL(F30_IN(x₁, x₂)) = [2] + x₁
POL(F32_IN(x₁)) = [1]
POL(F34_IN(x₁, x₂, x₃)) = 0
POL(F8_IN(x₁)) = [1] + [2]x₁
POL(F9_IN(x₁)) = [1] + x₁
POL(U12'(x₁, x₂, x₃)) = x₂
POL(U4'(x₁, x₂, x₃)) = [1] + x₂
POL(U7(x₁, x₂)) = 0
POL(U8'(x₁, x₂, x₃)) = [1] + x₂
POL([]) = 0
POL(c1(x₁)) = x₁
POL(c2) = 0
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4) = 0
POL(c4(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9) = 0
POL(f32_in(x₁)) = 0
POL(f32_out1(x₁, x₂)) = 0

(55) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
F9_IN(.(z0, z1)) → c4(F9_IN(z1))
U4'(f32_out1(z0, z1), z2, z3) → c9
F32_IN(.(z0, z1)) → c1(F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4
F34_IN(.(z0, z1), z2, z3) → c1(F34_IN(z1, z2, z3))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
U12'(f32_out1(z0, z1), z2, z3) → c2

S tuples:none
K tuples:

F2_IN(z0) → c2(F8_IN(z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U4'(f32_out1(z0, z1), z2, z3) → c9
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
U12'(f32_out1(z0, z1), z2, z3) → c2
U8'(f32_out1(z0, z1), z2, z3) → c4
F9_IN(.(z0, z1)) → c4(F9_IN(z1))

Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F30_IN, F2_IN, F10_IN, F8_IN, U12', F9_IN, U4', F32_IN, U8', F34_IN

Compound Symbols:

c8, c3, c2, c4, c9, c1, c4, c2

(56) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(57) BOUNDS(O(1), O(1))

(58) Obligation:

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)
Tuples:

F2_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c6(U3'(f30_in(z1, z0), .(z0, z1)), F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c11(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))
F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F30_IN(z0, z1) → c3(U8'(f32_in(z0), z0, z1), F32_IN(z0))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))
F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c4(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1), F34_IN(z1, z3, z0))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))
S tuples:

F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F30_IN(z0, z1) → c3(U8'(f32_in(z0), z0, z1), F32_IN(z0))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))
K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F2_IN, F9_IN, F10_IN, F30_IN, U4', F8_IN, F32_IN, U8', F34_IN, U12'

Compound Symbols:

c, c4, c6, c8, c9, c11, c1, c3, c4, c6, c3

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)
Tuples:

F2_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c6(U3'(f30_in(z1, z0), .(z0, z1)), F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c11(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))
F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F30_IN(z0, z1) → c3(U8'(f32_in(z0), z0, z1), F32_IN(z0))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))
F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c4(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1), F34_IN(z1, z3, z0))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))
S tuples:

F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c4(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1), F34_IN(z1, z3, z0))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))
K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F2_IN, F9_IN, F10_IN, F30_IN, U4', F8_IN, F32_IN, U8', F34_IN, U12'

Compound Symbols:

c, c4, c6, c8, c9, c11, c1, c3, c4, c6, c3

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

Multiplied the complexity of the problems.

(60) Complex Obligation (MULT)

(61) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F2_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c6(U3'(f30_in(z1, z0), .(z0, z1)), F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c11(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))
F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F30_IN(z0, z1) → c3(U8'(f32_in(z0), z0, z1), F32_IN(z0))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))
F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c4(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1), F34_IN(z1, z3, z0))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))

S tuples:

F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F30_IN(z0, z1) → c3(U8'(f32_in(z0), z0, z1), F32_IN(z0))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F2_IN, F9_IN, F10_IN, F30_IN, U4', F8_IN, F32_IN, U8', F34_IN, U12'

Compound Symbols:

c, c4, c6, c8, c9, c11, c1, c3, c4, c6, c3

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

Split RHS of tuples not part of any SCC

(63) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))
F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))
F2_IN(z0) → c2(U1'(f8_in(z0), z0))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(U3'(f30_in(z1, z0), .(z0, z1)))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(U6'(f9_in(z0), f10_in(z0), z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))

S tuples:

F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F9_IN, F30_IN, U4', F32_IN, U8', F8_IN, F34_IN, F2_IN, F10_IN, U12'

Compound Symbols:

c4, c8, c9, c1, c4, c6, c3, c2

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

Removed 11 trailing tuple parts

(65) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
F9_IN(.(z0, z1)) → c4(F9_IN(z1))
U4'(f32_out1(z0, z1), z2, z3) → c9
F32_IN(.(z0, z1)) → c1(F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4
F8_IN(z0) → c6
F34_IN(.(z0, z1), z2, z3) → c1(F34_IN(z1, z2, z3))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
U12'(f32_out1(z0, z1), z2, z3) → c2

S tuples:

F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
F32_IN(.(z0, z1)) → c1(F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4
F8_IN(z0) → c6

K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F30_IN, F2_IN, F10_IN, F8_IN, U12', F9_IN, U4', F32_IN, U8', F34_IN

Compound Symbols:

c8, c3, c2, c4, c9, c1, c4, c6, 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.

F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
F8_IN(z0) → c6

We considered the (Usable) Rules:

f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)

And the Tuples:

F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
F9_IN(.(z0, z1)) → c4(F9_IN(z1))
U4'(f32_out1(z0, z1), z2, z3) → c9
F32_IN(.(z0, z1)) → c1(F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4
F8_IN(z0) → c6
F34_IN(.(z0, z1), z2, z3) → c1(F34_IN(z1, z2, z3))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
U12'(f32_out1(z0, z1), z2, z3) → c2

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

POL(.(x₁, x₂)) = [2] + x₂
POL(F10_IN(x₁)) = [3]
POL(F2_IN(x₁)) = [3] + [2]x₁
POL(F30_IN(x₁, x₂)) = [2]
POL(F32_IN(x₁)) = [1]
POL(F34_IN(x₁, x₂, x₃)) = 0
POL(F8_IN(x₁)) = [3] + x₁
POL(F9_IN(x₁)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U4'(x₁, x₂, x₃)) = [1]
POL(U7(x₁, x₂)) = x₂
POL(U8'(x₁, x₂, x₃)) = 0
POL([]) = 0
POL(c1(x₁)) = x₁
POL(c2) = 0
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4) = 0
POL(c4(x₁)) = x₁
POL(c6) = 0
POL(c8(x₁)) = x₁
POL(c9) = 0
POL(f32_in(x₁)) = [2]x₁
POL(f32_out1(x₁, x₂)) = 0

(67) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
F9_IN(.(z0, z1)) → c4(F9_IN(z1))
U4'(f32_out1(z0, z1), z2, z3) → c9
F32_IN(.(z0, z1)) → c1(F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4
F8_IN(z0) → c6
F34_IN(.(z0, z1), z2, z3) → c1(F34_IN(z1, z2, z3))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
U12'(f32_out1(z0, z1), z2, z3) → c2

S tuples:

F32_IN(.(z0, z1)) → c1(F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4

K tuples:

F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
F8_IN(z0) → c6

Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F30_IN, F2_IN, F10_IN, F8_IN, U12', F9_IN, U4', F32_IN, U8', F34_IN

Compound Symbols:

c8, c3, c2, c4, c9, c1, c4, c6, c2

(68) CdtKnowledgeProof (EQUIVALENT transformation)

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

U8'(f32_out1(z0, z1), z2, z3) → c4

(69) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
F9_IN(.(z0, z1)) → c4(F9_IN(z1))
U4'(f32_out1(z0, z1), z2, z3) → c9
F32_IN(.(z0, z1)) → c1(F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4
F8_IN(z0) → c6
F34_IN(.(z0, z1), z2, z3) → c1(F34_IN(z1, z2, z3))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
U12'(f32_out1(z0, z1), z2, z3) → c2

S tuples:

F32_IN(.(z0, z1)) → c1(F32_IN(z1))

K tuples:

F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
F8_IN(z0) → c6
U8'(f32_out1(z0, z1), z2, z3) → c4

Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F30_IN, F2_IN, F10_IN, F8_IN, U12', F9_IN, U4', F32_IN, U8', F34_IN

Compound Symbols:

c8, c3, c2, c4, c9, c1, c4, c6, c2

(70) 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.

F32_IN(.(z0, z1)) → c1(F32_IN(z1))

We considered the (Usable) Rules:

f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)

And the Tuples:

F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
F9_IN(.(z0, z1)) → c4(F9_IN(z1))
U4'(f32_out1(z0, z1), z2, z3) → c9
F32_IN(.(z0, z1)) → c1(F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4
F8_IN(z0) → c6
F34_IN(.(z0, z1), z2, z3) → c1(F34_IN(z1, z2, z3))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
U12'(f32_out1(z0, z1), z2, z3) → c2

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

POL(.(x₁, x₂)) = [2] + x₂
POL(F10_IN(x₁)) = [2]x₁
POL(F2_IN(x₁)) = [3]x₁
POL(F30_IN(x₁, x₂)) = [3] + x₁
POL(F32_IN(x₁)) = x₁
POL(F34_IN(x₁, x₂, x₃)) = 0
POL(F8_IN(x₁)) = [3]x₁
POL(F9_IN(x₁)) = 0
POL(U12'(x₁, x₂, x₃)) = x₂
POL(U4'(x₁, x₂, x₃)) = [2]
POL(U7(x₁, x₂)) = 0
POL(U8'(x₁, x₂, x₃)) = [2] + x₂
POL([]) = 0
POL(c1(x₁)) = x₁
POL(c2) = 0
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4) = 0
POL(c4(x₁)) = x₁
POL(c6) = 0
POL(c8(x₁)) = x₁
POL(c9) = 0
POL(f32_in(x₁)) = 0
POL(f32_out1(x₁, x₂)) = 0

(71) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
F9_IN(.(z0, z1)) → c4(F9_IN(z1))
U4'(f32_out1(z0, z1), z2, z3) → c9
F32_IN(.(z0, z1)) → c1(F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4
F8_IN(z0) → c6
F34_IN(.(z0, z1), z2, z3) → c1(F34_IN(z1, z2, z3))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
U12'(f32_out1(z0, z1), z2, z3) → c2

S tuples:none
K tuples:

F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
F8_IN(z0) → c6
U8'(f32_out1(z0, z1), z2, z3) → c4
F32_IN(.(z0, z1)) → c1(F32_IN(z1))

Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F30_IN, F2_IN, F10_IN, F8_IN, U12', F9_IN, U4', F32_IN, U8', F34_IN

Compound Symbols:

c8, c3, c2, c4, c9, c1, c4, c6, c2

(72) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(73) BOUNDS(O(1), O(1))

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F2_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c6(U3'(f30_in(z1, z0), .(z0, z1)), F30_IN(z1, z0))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c11(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))
F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F30_IN(z0, z1) → c3(U8'(f32_in(z0), z0, z1), F32_IN(z0))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))
F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c4(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1), F34_IN(z1, z3, z0))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))

S tuples:

F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c4(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1), F34_IN(z1, z3, z0))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F2_IN, F9_IN, F10_IN, F30_IN, U4', F8_IN, F32_IN, U8', F34_IN, U12'

Compound Symbols:

c, c4, c6, c8, c9, c11, c1, c3, c4, c6, c3

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

Split RHS of tuples not part of any SCC

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F9_IN(.(z0, z1)) → c4(U2'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
U4'(f32_out1(z0, z1), z2, z3) → c9(U5'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F32_IN(.(z0, z1)) → c1(U7'(f32_in(z1), .(z0, z1)), F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4(U9'(f34_in(z1, z3, z0), z2, z3, z0, z1))
F8_IN(z0) → c6(U10'(f9_in(z0), f10_in(z0), z0))
F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))
F2_IN(z0) → c2(U1'(f8_in(z0), z0))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(U3'(f30_in(z1, z0), .(z0, z1)))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(U6'(f9_in(z0), f10_in(z0), z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))

S tuples:

F34_IN(.(z0, z1), z2, z3) → c1(U11'(f34_in(z1, z2, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z3))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F8_IN(z0) → c6(U14'(f9_in(z0), f10_in(z0), z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(U13'(f34_in(z1, z3, z0), z2, z3, z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F9_IN, F30_IN, U4', F32_IN, U8', F8_IN, F34_IN, F2_IN, F10_IN, U12'

Compound Symbols:

c4, c8, c9, c1, c4, c6, c3, c2

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

Removed 11 trailing tuple parts

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0), z1) → f2_out1(z0)
U1(f8_out2(z0, z1, z2), z3) → f2_out1(z2)
f9_in([]) → f9_out1([])
f9_in(.(z0, z1)) → U2(f9_in(z1), .(z0, z1))
U2(f9_out1(z0), .(z1, z2)) → f9_out1(.(z1, z0))
f10_in(.(z0, z1)) → U3(f30_in(z1, z0), .(z0, z1))
U3(f30_out1(z0, z1, z2), .(z3, z4)) → f10_out1(.(z3, z0), z1, z2)
f30_in(z0, z1) → U4(f32_in(z0), z0, z1)
f30_in(z0, z1) → U8(f32_in(z0), z0, z1)
f30_in(z0, z1) → U12(f32_in(z0), z0, z1)
U4(f32_out1(z0, z1), z2, z3) → U5(f34_in(z1, z3, z0), z2, z3, z0, z1)
U5(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
f8_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U10(f9_in(z0), f10_in(z0), z0)
f8_in(z0) → U14(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0), z1, z2) → f8_out1(z0)
U6(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f32_in(z0) → f32_out1([], z0)
f32_in(.(z0, z1)) → U7(f32_in(z1), .(z0, z1))
U7(f32_out1(z0, z1), .(z2, z3)) → f32_out1(.(z2, z0), z1)
U8(f32_out1(z0, z1), z2, z3) → U9(f34_in(z1, z3, z0), z2, z3, z0, z1)
U9(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U10(f9_out1(z0), z1, z2) → f8_out1(z0)
U10(z0, f10_out1(z1, z2, z3), z4) → f8_out2(z1, z2, z3)
f34_in([], z0, z1) → f34_out1(.(z0, z1))
f34_in(.(z0, z1), z2, z3) → U11(f34_in(z1, z2, z3), .(z0, z1), z2, z3)
U11(f34_out1(z0), .(z1, z2), z3, z4) → f34_out1(.(z1, z0))
U12(f32_out1(z0, z1), z2, z3) → U13(f34_in(z1, z3, z0), z2, z3, z0, z1)
U13(f34_out1(z0), z1, z2, z3, z4) → f30_out1(z3, z4, z0)
U14(f9_out1(z0), z1, z2) → f8_out1(z0)
U14(z0, f10_out1(z1, z0, z2), z3) → f8_out2(z1, z0, z2)

Tuples:

F30_IN(z0, z1) → c8(U4'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
F2_IN(z0) → c2(F8_IN(z0))
F10_IN(.(z0, z1)) → c2(F30_IN(z1, z0))
F8_IN(z0) → c2(F9_IN(z0))
F8_IN(z0) → c2(F10_IN(z0))
F30_IN(z0, z1) → c2(U8'(f32_in(z0), z0, z1))
F30_IN(z0, z1) → c2(F32_IN(z0))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
F9_IN(.(z0, z1)) → c4(F9_IN(z1))
U4'(f32_out1(z0, z1), z2, z3) → c9
F32_IN(.(z0, z1)) → c1(F32_IN(z1))
U8'(f32_out1(z0, z1), z2, z3) → c4
F8_IN(z0) → c6
F34_IN(.(z0, z1), z2, z3) → c1(F34_IN(z1, z2, z3))
F2_IN(z0) → c2
F10_IN(.(z0, z1)) → c2
F8_IN(z0) → c2
U12'(f32_out1(z0, z1), z2, z3) → c2

S tuples:

F30_IN(z0, z1) → c3(U12'(f32_in(z0), z0, z1))
U12'(f32_out1(z0, z1), z2, z3) → c2(F34_IN(z1, z3, z0))
F34_IN(.(z0, z1), z2, z3) → c1(F34_IN(z1, z2, z3))
F8_IN(z0) → c6
U12'(f32_out1(z0, z1), z2, z3) → c2

K tuples:none
Defined Rule Symbols:

f2_in, U1, f9_in, U2, f10_in, U3, f30_in, U4, U5, f8_in, U6, f32_in, U7, U8, U9, U10, f34_in, U11, U12, U13, U14

Defined Pair Symbols:

F30_IN, F2_IN, F10_IN, F8_IN, U12', F9_IN, U4', F32_IN, U8', F34_IN

Compound Symbols:

c8, c3, c2, c4, c9, c1, c4, c6, c2