Runtime Complexity proof of /tmp/tmpAi7WUl/sublist-fb.pl

Runtime Complexity of the query pattern
sublist(a,g)
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), 71 ms)

↳4 CdtProblem

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

    ↳6 CdtProblem

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

    ↳8 CdtProblem

    ↳9 CdtKnowledgeProof (⇔, 2 ms)

    ↳10 CdtProblem

    ↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 177 ms)

    ↳12 CdtProblem

    ↳13 CdtKnowledgeProof (⇔, 0 ms)

    ↳14 CdtProblem

    ↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 0 ms)

    ↳16 CdtProblem

    ↳17 SIsEmptyProof (⇔, 0 ms)

    ↳18 BOUNDS(O(1), O(1))

↳19 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 141 ms)

↳20 ComplexCdtProblem

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

    ↳22 MAX

        ↳23 CdtProblem

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

            ↳25 CdtProblem

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

            ↳27 CdtProblem

            ↳28 CdtKnowledgeProof (⇔, 0 ms)

            ↳29 BOUNDS(O(1), O(1))

        ↳30 ComplexCdtProblem

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

            ↳32 MULT

                ↳33 CdtProblem

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

                    ↳35 CdtProblem

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

                    ↳37 CdtProblem

                    ↳38 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 143 ms)

                    ↳39 CdtProblem

                ↳40 CdtProblem

(0) Obligation:

Clauses:

sublist(Xs, Ys) :- ','(app(X1, Zs, Ys), app(Xs, X2, Zs)).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Query: sublist(a,g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

app([], X, X).
sublist(Xs, Ys) :- ','(app(X1, Zs, Ys), app(Xs, X2, Zs)).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Query: sublist(a,g)

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

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: sublist(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: sublist(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(app(X7, X8, T6), app(T7, X9, X8))\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(app(X7, X8, T6), app(T7, X9, X8)), SCOPE: 2, CLAUSE: 0 | ','(app(X7, X8, T6), app(T7, X9, X8)), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: app(T7, X9, T12) | ','(app(X7, X8, T12), app(T7, X9, X8)), SCOPE: 2, CLAUSE: 2\n\nT12 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: app(T7, X9, T12)\n\nT12 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(app(X7, X8, T12), app(T7, X9, X8)), SCOPE: 2, CLAUSE: 2\n\nT12 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: app(T7, X9, T12), SCOPE: 3, CLAUSE: 0 | app(T7, X9, T12), SCOPE: 3, CLAUSE: 2\n\nT12 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true | app(T7, X9, T19), SCOPE: 3, CLAUSE: 2\n\nT19 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: app(T7, X9, T12), SCOPE: 3, CLAUSE: 2\n\nT12 is ground\nX9 is free\nX32 is free\napp(T7, X9, T12) !=? app([], X32, X32)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: app(T7, X9, T19), SCOPE: 3, CLAUSE: 2\n\nT19 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(T29, X48, T28)\n\nT28 is ground\nX48 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: app(T41, X67, T40)\n\nT38 is ground\nT40 is ground\nX9 is free\nX32 is free\nX67 is free\napp(T7, X9, .(T38, T40)) !=? app([], X32, X32)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(app(X91, X92, T49), app(T7, X9, X92))\n\nT49 is ground\nX9 is free\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 34 [arrowhead = none ];
34 -> 4;

35 [label="ONLY EVAL with clause\nsublist(X5, X6) :- ','(app(X7, X8, X6), app(X5, X9, X8)).\nand substitutionT1 -> T7,\nX5 -> T7,\nT2 -> T6,\nX6 -> T6,\nT5 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 35 [arrowhead = none ];
35 -> 5;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 36 [arrowhead = none ];
36 -> 6;

37 [label="ONLY EVAL with clause\napp([], X18, X18).\nand substitutionX7 -> [],\nX8 -> T12,\nX18 -> T12,\nT6 -> T12,\nX19 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 37 [arrowhead = none ];
37 -> 7;

38 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 38 [arrowhead = none ];
38 -> 9;

39 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 39 [arrowhead = none ];
39 -> 10;

40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 40 [arrowhead = none ];
40 -> 13;

41 [label="EVAL with clause\napp(.(X86, X87), X88, .(X86, X89)) :- app(X87, X88, X89).\nand substitutionX86 -> T48,\nX87 -> X91,\nX7 -> .(T48, X91),\nX8 -> X92,\nX88 -> X92,\nX90 -> T48,\nX89 -> T49,\nT12 -> .(T48, T49)", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 41 [arrowhead = none ];
41 -> 32;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 42 [arrowhead = none ];
42 -> 33;

43 [label="EVAL with clause\napp([], X32, X32).\nand substitutionT7 -> [],\nX9 -> T19,\nX32 -> T19,\nT12 -> T19,\nX33 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 43 [arrowhead = none ];
43 -> 14;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 44 [arrowhead = none ];
44 -> 15;

45 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 45 [arrowhead = none ];
45 -> 17;

46 [label="EVAL with clause\napp(.(X63, X64), X65, .(X63, X66)) :- app(X64, X65, X66).\nand substitutionX63 -> T38,\nX64 -> T41,\nT7 -> .(T38, T41),\nX9 -> X67,\nX65 -> X67,\nX66 -> T40,\nT12 -> .(T38, T40),\nT39 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 46 [arrowhead = none ];
46 -> 28;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 47 [arrowhead = none ];
47 -> 29;

48 [label="EVAL with clause\napp(.(X44, X45), X46, .(X44, X47)) :- app(X45, X46, X47).\nand substitutionX44 -> T26,\nX45 -> T29,\nT7 -> .(T26, T29),\nX9 -> X48,\nX46 -> X48,\nX47 -> T28,\nT19 -> .(T26, T28),\nT27 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 48 [arrowhead = none ];
48 -> 20;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 49 [arrowhead = none ];
49 -> 21;

50 [label="INSTANCE with matching:\nT7 -> T29\nX9 -> X48\nT12 -> T28", fontsize=14, style = filled, fillcolor = lightgrey];
20 -> 50 [arrowhead = none , style=dashed];
50 -> 9[style=dashed];

51 [label="INSTANCE with matching:\nT7 -> T41\nX9 -> X67\nT12 -> T40", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 51 [arrowhead = none , style=dashed];
51 -> 9[style=dashed];

52 [label="INSTANCE with matching:\nX7 -> X91\nX8 -> X92\nT6 -> T49", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 52 [arrowhead = none , style=dashed];
52 -> 5[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1(z0, z1, z2, z3), z4) → f2_out1(z2)
f5_in(z0) → U2(f7_in(z0), z0)
U2(f7_out1(z0, z1), z2) → f5_out1([], z2, z0, z1)
U2(f7_out2(z0, z1, z2, z3), z4) → f5_out1(z0, z1, z2, z3)
f9_in(z0) → f9_out1([], z0)
f9_in(.(z0, z1)) → U3(f9_in(z1), .(z0, z1))
f9_in(.(z0, z1)) → U4(f9_in(z1), .(z0, z1))
U3(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
U4(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
f10_in(.(z0, z1)) → U5(f5_in(z1), .(z0, z1))
U5(f5_out1(z0, z1, z2, z3), .(z4, z5)) → f10_out1(.(z4, z0), z1, z2, z3)
f7_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0, z1), z2, z3) → f7_out1(z0, z1)
U6(z0, f10_out1(z1, z2, z3, z4), z5) → f7_out2(z1, z2, z3, z4)

Tuples:

F2_IN(z0) → c(U1'(f5_in(z0), z0), F5_IN(z0))
F5_IN(z0) → c2(U2'(f7_in(z0), z0), F7_IN(z0))
F9_IN(.(z0, z1)) → c6(U3'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F9_IN(.(z0, z1)) → c7(U4'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c10(U5'(f5_in(z1), .(z0, z1)), F5_IN(z1))
F7_IN(z0) → c12(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))

S tuples:

F2_IN(z0) → c(U1'(f5_in(z0), z0), F5_IN(z0))
F5_IN(z0) → c2(U2'(f7_in(z0), z0), F7_IN(z0))
F9_IN(.(z0, z1)) → c6(U3'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F9_IN(.(z0, z1)) → c7(U4'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c10(U5'(f5_in(z1), .(z0, z1)), F5_IN(z1))
F7_IN(z0) → c12(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f5_in, U2, f9_in, U3, U4, f10_in, U5, f7_in, U6

Defined Pair Symbols:

F2_IN, F5_IN, F9_IN, F10_IN, F7_IN

Compound Symbols:

c, c2, c6, c7, c10, c12

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

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1(z0, z1, z2, z3), z4) → f2_out1(z2)
f5_in(z0) → U2(f7_in(z0), z0)
U2(f7_out1(z0, z1), z2) → f5_out1([], z2, z0, z1)
U2(f7_out2(z0, z1, z2, z3), z4) → f5_out1(z0, z1, z2, z3)
f9_in(z0) → f9_out1([], z0)
f9_in(.(z0, z1)) → U3(f9_in(z1), .(z0, z1))
f9_in(.(z0, z1)) → U4(f9_in(z1), .(z0, z1))
U3(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
U4(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
f10_in(.(z0, z1)) → U5(f5_in(z1), .(z0, z1))
U5(f5_out1(z0, z1, z2, z3), .(z4, z5)) → f10_out1(.(z4, z0), z1, z2, z3)
f7_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0, z1), z2, z3) → f7_out1(z0, z1)
U6(z0, f10_out1(z1, z2, z3, z4), z5) → f7_out2(z1, z2, z3, z4)

Tuples:

F5_IN(z0) → c2(U2'(f7_in(z0), z0), F7_IN(z0))
F9_IN(.(z0, z1)) → c6(U3'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F9_IN(.(z0, z1)) → c7(U4'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c10(U5'(f5_in(z1), .(z0, z1)), F5_IN(z1))
F7_IN(z0) → c12(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))
F2_IN(z0) → c1(U1'(f5_in(z0), z0))
F2_IN(z0) → c1(F5_IN(z0))

S tuples:

F5_IN(z0) → c2(U2'(f7_in(z0), z0), F7_IN(z0))
F9_IN(.(z0, z1)) → c6(U3'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F9_IN(.(z0, z1)) → c7(U4'(f9_in(z1), .(z0, z1)), F9_IN(z1))
F10_IN(.(z0, z1)) → c10(U5'(f5_in(z1), .(z0, z1)), F5_IN(z1))
F7_IN(z0) → c12(U6'(f9_in(z0), f10_in(z0), z0), F9_IN(z0), F10_IN(z0))
F2_IN(z0) → c1(U1'(f5_in(z0), z0))
F2_IN(z0) → c1(F5_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f5_in, U2, f9_in, U3, U4, f10_in, U5, f7_in, U6

Defined Pair Symbols:

F5_IN, F9_IN, F10_IN, F7_IN, F2_IN

Compound Symbols:

c2, c6, c7, c10, c12, c1

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

Removed 6 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1(z0, z1, z2, z3), z4) → f2_out1(z2)
f5_in(z0) → U2(f7_in(z0), z0)
U2(f7_out1(z0, z1), z2) → f5_out1([], z2, z0, z1)
U2(f7_out2(z0, z1, z2, z3), z4) → f5_out1(z0, z1, z2, z3)
f9_in(z0) → f9_out1([], z0)
f9_in(.(z0, z1)) → U3(f9_in(z1), .(z0, z1))
f9_in(.(z0, z1)) → U4(f9_in(z1), .(z0, z1))
U3(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
U4(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
f10_in(.(z0, z1)) → U5(f5_in(z1), .(z0, z1))
U5(f5_out1(z0, z1, z2, z3), .(z4, z5)) → f10_out1(.(z4, z0), z1, z2, z3)
f7_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0, z1), z2, z3) → f7_out1(z0, z1)
U6(z0, f10_out1(z1, z2, z3, z4), z5) → f7_out2(z1, z2, z3, z4)

Tuples:

F2_IN(z0) → c1(F5_IN(z0))
F5_IN(z0) → c2(F7_IN(z0))
F9_IN(.(z0, z1)) → c6(F9_IN(z1))
F9_IN(.(z0, z1)) → c7(F9_IN(z1))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))
F7_IN(z0) → c12(F9_IN(z0), F10_IN(z0))
F2_IN(z0) → c1

S tuples:

F2_IN(z0) → c1(F5_IN(z0))
F5_IN(z0) → c2(F7_IN(z0))
F9_IN(.(z0, z1)) → c6(F9_IN(z1))
F9_IN(.(z0, z1)) → c7(F9_IN(z1))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))
F7_IN(z0) → c12(F9_IN(z0), F10_IN(z0))
F2_IN(z0) → c1

K tuples:none
Defined Rule Symbols:

f2_in, U1, f5_in, U2, f9_in, U3, U4, f10_in, U5, f7_in, U6

Defined Pair Symbols:

F2_IN, F5_IN, F9_IN, F10_IN, F7_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN(z0) → c1(F5_IN(z0))
F2_IN(z0) → c1

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1(z0, z1, z2, z3), z4) → f2_out1(z2)
f5_in(z0) → U2(f7_in(z0), z0)
U2(f7_out1(z0, z1), z2) → f5_out1([], z2, z0, z1)
U2(f7_out2(z0, z1, z2, z3), z4) → f5_out1(z0, z1, z2, z3)
f9_in(z0) → f9_out1([], z0)
f9_in(.(z0, z1)) → U3(f9_in(z1), .(z0, z1))
f9_in(.(z0, z1)) → U4(f9_in(z1), .(z0, z1))
U3(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
U4(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
f10_in(.(z0, z1)) → U5(f5_in(z1), .(z0, z1))
U5(f5_out1(z0, z1, z2, z3), .(z4, z5)) → f10_out1(.(z4, z0), z1, z2, z3)
f7_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0, z1), z2, z3) → f7_out1(z0, z1)
U6(z0, f10_out1(z1, z2, z3, z4), z5) → f7_out2(z1, z2, z3, z4)

Tuples:

F2_IN(z0) → c1(F5_IN(z0))
F5_IN(z0) → c2(F7_IN(z0))
F9_IN(.(z0, z1)) → c6(F9_IN(z1))
F9_IN(.(z0, z1)) → c7(F9_IN(z1))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))
F7_IN(z0) → c12(F9_IN(z0), F10_IN(z0))
F2_IN(z0) → c1

S tuples:

F5_IN(z0) → c2(F7_IN(z0))
F9_IN(.(z0, z1)) → c6(F9_IN(z1))
F9_IN(.(z0, z1)) → c7(F9_IN(z1))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))
F7_IN(z0) → c12(F9_IN(z0), F10_IN(z0))

K tuples:

F2_IN(z0) → c1(F5_IN(z0))
F2_IN(z0) → c1

Defined Rule Symbols:

f2_in, U1, f5_in, U2, f9_in, U3, U4, f10_in, U5, f7_in, U6

Defined Pair Symbols:

F2_IN, F5_IN, F9_IN, F10_IN, F7_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

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

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

F5_IN(z0) → c2(F7_IN(z0))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))

We considered the (Usable) Rules:none
And the Tuples:

F2_IN(z0) → c1(F5_IN(z0))
F5_IN(z0) → c2(F7_IN(z0))
F9_IN(.(z0, z1)) → c6(F9_IN(z1))
F9_IN(.(z0, z1)) → c7(F9_IN(z1))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))
F7_IN(z0) → c12(F9_IN(z0), F10_IN(z0))
F2_IN(z0) → c1

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)) = [2] + [2]x₁
POL(F5_IN(x₁)) = [1] + x₁
POL(F7_IN(x₁)) = x₁
POL(F9_IN(x₁)) = 0
POL(c1) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1(z0, z1, z2, z3), z4) → f2_out1(z2)
f5_in(z0) → U2(f7_in(z0), z0)
U2(f7_out1(z0, z1), z2) → f5_out1([], z2, z0, z1)
U2(f7_out2(z0, z1, z2, z3), z4) → f5_out1(z0, z1, z2, z3)
f9_in(z0) → f9_out1([], z0)
f9_in(.(z0, z1)) → U3(f9_in(z1), .(z0, z1))
f9_in(.(z0, z1)) → U4(f9_in(z1), .(z0, z1))
U3(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
U4(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
f10_in(.(z0, z1)) → U5(f5_in(z1), .(z0, z1))
U5(f5_out1(z0, z1, z2, z3), .(z4, z5)) → f10_out1(.(z4, z0), z1, z2, z3)
f7_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0, z1), z2, z3) → f7_out1(z0, z1)
U6(z0, f10_out1(z1, z2, z3, z4), z5) → f7_out2(z1, z2, z3, z4)

Tuples:

F2_IN(z0) → c1(F5_IN(z0))
F5_IN(z0) → c2(F7_IN(z0))
F9_IN(.(z0, z1)) → c6(F9_IN(z1))
F9_IN(.(z0, z1)) → c7(F9_IN(z1))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))
F7_IN(z0) → c12(F9_IN(z0), F10_IN(z0))
F2_IN(z0) → c1

S tuples:

F9_IN(.(z0, z1)) → c6(F9_IN(z1))
F9_IN(.(z0, z1)) → c7(F9_IN(z1))
F7_IN(z0) → c12(F9_IN(z0), F10_IN(z0))

K tuples:

F2_IN(z0) → c1(F5_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c2(F7_IN(z0))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))

Defined Rule Symbols:

f2_in, U1, f5_in, U2, f9_in, U3, U4, f10_in, U5, f7_in, U6

Defined Pair Symbols:

F2_IN, F5_IN, F9_IN, F10_IN, F7_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

(13) CdtKnowledgeProof (EQUIVALENT transformation)

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

F7_IN(z0) → c12(F9_IN(z0), F10_IN(z0))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1(z0, z1, z2, z3), z4) → f2_out1(z2)
f5_in(z0) → U2(f7_in(z0), z0)
U2(f7_out1(z0, z1), z2) → f5_out1([], z2, z0, z1)
U2(f7_out2(z0, z1, z2, z3), z4) → f5_out1(z0, z1, z2, z3)
f9_in(z0) → f9_out1([], z0)
f9_in(.(z0, z1)) → U3(f9_in(z1), .(z0, z1))
f9_in(.(z0, z1)) → U4(f9_in(z1), .(z0, z1))
U3(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
U4(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
f10_in(.(z0, z1)) → U5(f5_in(z1), .(z0, z1))
U5(f5_out1(z0, z1, z2, z3), .(z4, z5)) → f10_out1(.(z4, z0), z1, z2, z3)
f7_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0, z1), z2, z3) → f7_out1(z0, z1)
U6(z0, f10_out1(z1, z2, z3, z4), z5) → f7_out2(z1, z2, z3, z4)

Tuples:

F2_IN(z0) → c1(F5_IN(z0))
F5_IN(z0) → c2(F7_IN(z0))
F9_IN(.(z0, z1)) → c6(F9_IN(z1))
F9_IN(.(z0, z1)) → c7(F9_IN(z1))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))
F7_IN(z0) → c12(F9_IN(z0), F10_IN(z0))
F2_IN(z0) → c1

S tuples:

F9_IN(.(z0, z1)) → c6(F9_IN(z1))
F9_IN(.(z0, z1)) → c7(F9_IN(z1))

K tuples:

F2_IN(z0) → c1(F5_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c2(F7_IN(z0))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))
F7_IN(z0) → c12(F9_IN(z0), F10_IN(z0))

Defined Rule Symbols:

f2_in, U1, f5_in, U2, f9_in, U3, U4, f10_in, U5, f7_in, U6

Defined Pair Symbols:

F2_IN, F5_IN, F9_IN, F10_IN, F7_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

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

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

F9_IN(.(z0, z1)) → c6(F9_IN(z1))
F9_IN(.(z0, z1)) → c7(F9_IN(z1))

We considered the (Usable) Rules:none
And the Tuples:

F2_IN(z0) → c1(F5_IN(z0))
F5_IN(z0) → c2(F7_IN(z0))
F9_IN(.(z0, z1)) → c6(F9_IN(z1))
F9_IN(.(z0, z1)) → c7(F9_IN(z1))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))
F7_IN(z0) → c12(F9_IN(z0), F10_IN(z0))
F2_IN(z0) → c1

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

POL(.(x₁, x₂)) = [1] + x₂
POL(F10_IN(x₁)) = [1] + x₁²
POL(F2_IN(x₁)) = [1] + x₁ + x₁²
POL(F5_IN(x₁)) = [1] + x₁ + x₁²
POL(F7_IN(x₁)) = [1] + x₁ + x₁²
POL(F9_IN(x₁)) = x₁
POL(c1) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1(z0, z1, z2, z3), z4) → f2_out1(z2)
f5_in(z0) → U2(f7_in(z0), z0)
U2(f7_out1(z0, z1), z2) → f5_out1([], z2, z0, z1)
U2(f7_out2(z0, z1, z2, z3), z4) → f5_out1(z0, z1, z2, z3)
f9_in(z0) → f9_out1([], z0)
f9_in(.(z0, z1)) → U3(f9_in(z1), .(z0, z1))
f9_in(.(z0, z1)) → U4(f9_in(z1), .(z0, z1))
U3(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
U4(f9_out1(z0, z1), .(z2, z3)) → f9_out1(.(z2, z0), z1)
f10_in(.(z0, z1)) → U5(f5_in(z1), .(z0, z1))
U5(f5_out1(z0, z1, z2, z3), .(z4, z5)) → f10_out1(.(z4, z0), z1, z2, z3)
f7_in(z0) → U6(f9_in(z0), f10_in(z0), z0)
U6(f9_out1(z0, z1), z2, z3) → f7_out1(z0, z1)
U6(z0, f10_out1(z1, z2, z3, z4), z5) → f7_out2(z1, z2, z3, z4)

Tuples:

F2_IN(z0) → c1(F5_IN(z0))
F5_IN(z0) → c2(F7_IN(z0))
F9_IN(.(z0, z1)) → c6(F9_IN(z1))
F9_IN(.(z0, z1)) → c7(F9_IN(z1))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))
F7_IN(z0) → c12(F9_IN(z0), F10_IN(z0))
F2_IN(z0) → c1

S tuples:none
K tuples:

F2_IN(z0) → c1(F5_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c2(F7_IN(z0))
F10_IN(.(z0, z1)) → c10(F5_IN(z1))
F7_IN(z0) → c12(F9_IN(z0), F10_IN(z0))
F9_IN(.(z0, z1)) → c6(F9_IN(z1))
F9_IN(.(z0, z1)) → c7(F9_IN(z1))

Defined Rule Symbols:

f2_in, U1, f5_in, U2, f9_in, U3, U4, f10_in, U5, f7_in, U6

Defined Pair Symbols:

F2_IN, F5_IN, F9_IN, F10_IN, F7_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

(17) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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

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

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: sublist(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: sublist(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(app(X18, X19, T10), app(T11, X20, X19))\n\nT10 is ground\nX18 is free\nX19 is free\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: app(X18, X19, T10)\n\nT10 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: app(T11, X20, T14)\n\nT14 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: app(X18, X19, T10), SCOPE: 2, CLAUSE: 0 | app(X18, X19, T10), SCOPE: 2, CLAUSE: 2\n\nT10 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true | app(X18, X19, T18), SCOPE: 2, CLAUSE: 2\n\nT18 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: app(X18, X19, T18), SCOPE: 2, CLAUSE: 2\n\nT18 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app(X54, X55, T24)\n\nT24 is ground\nX54 is free\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: app(T11, X20, T14), SCOPE: 3, CLAUSE: 0 | app(T11, X20, T14), SCOPE: 3, CLAUSE: 2\n\nT14 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true | app(T11, X20, T30), SCOPE: 3, CLAUSE: 2\n\nT30 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: app(T11, X20, T14), SCOPE: 3, CLAUSE: 2\n\nT14 is ground\nX20 is free\nX66 is free\napp(T11, X20, T14) !=? app([], X66, X66)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app(T11, X20, T30), SCOPE: 3, CLAUSE: 2\n\nT30 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: app(T40, X82, T39)\n\nT39 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: app(T52, X101, T51)\n\nT49 is ground\nT51 is ground\nX20 is free\nX66 is free\nX101 is free\napp(T11, X20, .(T49, T51)) !=? app([], X66, X66)", 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 -> 3;

37 [label="ONLY EVAL with clause\nsublist(X16, X17) :- ','(app(X18, X19, X17), app(X16, X20, X19)).\nand substitutionT1 -> T11,\nX16 -> T11,\nT2 -> T10,\nX17 -> T10,\nT9 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 37 [arrowhead = none ];
37 -> 8;

38 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 38 [arrowhead = none ];
38 -> 11;

39 [label="SPLIT 2\nnew knowledge:\nT13 is ground\nT14 is ground\nT10 is ground\nreplacements:X18 -> T13,\nX19 -> T14", fontsize=14, style = filled, fillcolor = violet];
8 -> 39 [arrowhead = none ];
39 -> 12;

40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 40 [arrowhead = none ];
40 -> 16;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 41 [arrowhead = none ];
41 -> 24;

42 [label="ONLY EVAL with clause\napp([], X33, X33).\nand substitutionX18 -> [],\nX19 -> T18,\nX33 -> T18,\nT10 -> T18,\nX34 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 42 [arrowhead = none ];
42 -> 18;

43 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 43 [arrowhead = none ];
43 -> 19;

44 [label="EVAL with clause\napp(.(X49, X50), X51, .(X49, X52)) :- app(X50, X51, X52).\nand substitutionX49 -> T23,\nX50 -> X54,\nX18 -> .(T23, X54),\nX19 -> X55,\nX51 -> X55,\nX53 -> T23,\nX52 -> T24,\nT18 -> .(T23, T24)", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 44 [arrowhead = none ];
44 -> 22;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 45 [arrowhead = none ];
45 -> 23;

46 [label="INSTANCE with matching:\nX18 -> X54\nX19 -> X55\nT10 -> T24", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 46 [arrowhead = none , style=dashed];
46 -> 11[style=dashed];

47 [label="EVAL with clause\napp([], X66, X66).\nand substitutionT11 -> [],\nX20 -> T30,\nX66 -> T30,\nT14 -> T30,\nX67 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 47 [arrowhead = none ];
47 -> 25;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 48 [arrowhead = none ];
48 -> 26;

49 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 49 [arrowhead = none ];
49 -> 27;

50 [label="EVAL with clause\napp(.(X97, X98), X99, .(X97, X100)) :- app(X98, X99, X100).\nand substitutionX97 -> T49,\nX98 -> T52,\nT11 -> .(T49, T52),\nX20 -> X101,\nX99 -> X101,\nX100 -> T51,\nT14 -> .(T49, T51),\nT50 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 50 [arrowhead = none ];
50 -> 34;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 51 [arrowhead = none ];
51 -> 35;

52 [label="EVAL with clause\napp(.(X78, X79), X80, .(X78, X81)) :- app(X79, X80, X81).\nand substitutionX78 -> T37,\nX79 -> T40,\nT11 -> .(T37, T40),\nX20 -> X82,\nX80 -> X82,\nX81 -> T39,\nT30 -> .(T37, T39),\nT38 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 52 [arrowhead = none ];
52 -> 30;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 53 [arrowhead = none ];
53 -> 31;

54 [label="INSTANCE with matching:\nT11 -> T40\nX20 -> X82\nT14 -> T39", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 54 [arrowhead = none , style=dashed];
54 -> 12[style=dashed];

55 [label="INSTANCE with matching:\nT11 -> T52\nX20 -> X101\nT14 -> T51", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 55 [arrowhead = none , style=dashed];
55 -> 12[style=dashed];

}

(20) Obligation:

Complex Complexity Dependency Tuples Problem
MAX

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0, z1, z2, z3), z4) → f1_out1(z2)
f8_in(z0) → U2(f11_in(z0), z0)
f8_in(z0) → U5(f11_in(z0), z0)
f8_in(z0) → U9(f11_in(z0), z0)
U2(f11_out1(z0, z1), z2) → U3(f12_in(z1), z2, z0, z1)
U3(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2) → U6(f12_in(z1), z2, z0, z1)
U6(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U7(f12_in(z1), .(z0, z1))
f12_in(.(z0, z1)) → U8(f12_in(z1), .(z0, z1))
U7(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U8(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U9(f11_out1(z0, z1), z2) → U10(f12_in(z1), z2, z0, z1)
U10(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
Tuples:

F1_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F8_IN(z0) → c2(U2'(f11_in(z0), z0))
U2'(f11_out1(z0, z1), z2) → c3(U3'(f12_in(z1), z2, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F8_IN(z0) → c3(U5'(f11_in(z0), z0), F11_IN(z0))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U7'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F12_IN(.(z0, z1)) → c2(U8'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
U9'(f11_out1(z0, z1), z2) → c6(U10'(f12_in(z1), z2, z0, z1), F12_IN(z1))
S tuples:

F1_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F8_IN(z0) → c2(U2'(f11_in(z0), z0))
U2'(f11_out1(z0, z1), z2) → c3(U3'(f12_in(z1), z2, z0, z1))
K tuples:none
Defined Rule Symbols:

f1_in, U1, f8_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10

Defined Pair Symbols:

F1_IN, F8_IN, U2', F11_IN, U5', F12_IN, U9'

Compound Symbols:

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

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0, z1, z2, z3), z4) → f1_out1(z2)
f8_in(z0) → U2(f11_in(z0), z0)
f8_in(z0) → U5(f11_in(z0), z0)
f8_in(z0) → U9(f11_in(z0), z0)
U2(f11_out1(z0, z1), z2) → U3(f12_in(z1), z2, z0, z1)
U3(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2) → U6(f12_in(z1), z2, z0, z1)
U6(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U7(f12_in(z1), .(z0, z1))
f12_in(.(z0, z1)) → U8(f12_in(z1), .(z0, z1))
U7(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U8(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U9(f11_out1(z0, z1), z2) → U10(f12_in(z1), z2, z0, z1)
U10(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
Tuples:

F1_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F8_IN(z0) → c2(U2'(f11_in(z0), z0))
U2'(f11_out1(z0, z1), z2) → c3(U3'(f12_in(z1), z2, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F8_IN(z0) → c3(U5'(f11_in(z0), z0), F11_IN(z0))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U7'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F12_IN(.(z0, z1)) → c2(U8'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
U9'(f11_out1(z0, z1), z2) → c6(U10'(f12_in(z1), z2, z0, z1), F12_IN(z1))
S tuples:

F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F8_IN(z0) → c3(U5'(f11_in(z0), z0), F11_IN(z0))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))
K tuples:none
Defined Rule Symbols:

f1_in, U1, f8_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10

Defined Pair Symbols:

F1_IN, F8_IN, U2', F11_IN, U5', F12_IN, U9'

Compound Symbols:

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

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0, z1, z2, z3), z4) → f1_out1(z2)
f8_in(z0) → U2(f11_in(z0), z0)
f8_in(z0) → U5(f11_in(z0), z0)
f8_in(z0) → U9(f11_in(z0), z0)
U2(f11_out1(z0, z1), z2) → U3(f12_in(z1), z2, z0, z1)
U3(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2) → U6(f12_in(z1), z2, z0, z1)
U6(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U7(f12_in(z1), .(z0, z1))
f12_in(.(z0, z1)) → U8(f12_in(z1), .(z0, z1))
U7(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U8(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U9(f11_out1(z0, z1), z2) → U10(f12_in(z1), z2, z0, z1)
U10(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
Tuples:

F1_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F8_IN(z0) → c2(U2'(f11_in(z0), z0))
U2'(f11_out1(z0, z1), z2) → c3(U3'(f12_in(z1), z2, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F8_IN(z0) → c3(U5'(f11_in(z0), z0), F11_IN(z0))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U7'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F12_IN(.(z0, z1)) → c2(U8'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
U9'(f11_out1(z0, z1), z2) → c6(U10'(f12_in(z1), z2, z0, z1), F12_IN(z1))
S tuples:

F12_IN(.(z0, z1)) → c1(U7'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F12_IN(.(z0, z1)) → c2(U8'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
U9'(f11_out1(z0, z1), z2) → c6(U10'(f12_in(z1), z2, z0, z1), F12_IN(z1))
K tuples:none
Defined Rule Symbols:

f1_in, U1, f8_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10

Defined Pair Symbols:

F1_IN, F8_IN, U2', F11_IN, U5', F12_IN, U9'

Compound Symbols:

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

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

Took the maximum complexity of the problems.

(22) Complex Obligation (MAX)

(23) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0, z1, z2, z3), z4) → f1_out1(z2)
f8_in(z0) → U2(f11_in(z0), z0)
f8_in(z0) → U5(f11_in(z0), z0)
f8_in(z0) → U9(f11_in(z0), z0)
U2(f11_out1(z0, z1), z2) → U3(f12_in(z1), z2, z0, z1)
U3(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2) → U6(f12_in(z1), z2, z0, z1)
U6(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U7(f12_in(z1), .(z0, z1))
f12_in(.(z0, z1)) → U8(f12_in(z1), .(z0, z1))
U7(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U8(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U9(f11_out1(z0, z1), z2) → U10(f12_in(z1), z2, z0, z1)
U10(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)

Tuples:

F1_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F8_IN(z0) → c2(U2'(f11_in(z0), z0))
U2'(f11_out1(z0, z1), z2) → c3(U3'(f12_in(z1), z2, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F8_IN(z0) → c3(U5'(f11_in(z0), z0), F11_IN(z0))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U7'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F12_IN(.(z0, z1)) → c2(U8'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
U9'(f11_out1(z0, z1), z2) → c6(U10'(f12_in(z1), z2, z0, z1), F12_IN(z1))

S tuples:

F1_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F8_IN(z0) → c2(U2'(f11_in(z0), z0))
U2'(f11_out1(z0, z1), z2) → c3(U3'(f12_in(z1), z2, z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f8_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10

Defined Pair Symbols:

F1_IN, F8_IN, U2', F11_IN, U5', F12_IN, U9'

Compound Symbols:

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

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

Split RHS of tuples not part of any SCC

(25) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0, z1, z2, z3), z4) → f1_out1(z2)
f8_in(z0) → U2(f11_in(z0), z0)
f8_in(z0) → U5(f11_in(z0), z0)
f8_in(z0) → U9(f11_in(z0), z0)
U2(f11_out1(z0, z1), z2) → U3(f12_in(z1), z2, z0, z1)
U3(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2) → U6(f12_in(z1), z2, z0, z1)
U6(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U7(f12_in(z1), .(z0, z1))
f12_in(.(z0, z1)) → U8(f12_in(z1), .(z0, z1))
U7(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U8(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U9(f11_out1(z0, z1), z2) → U10(f12_in(z1), z2, z0, z1)
U10(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)

Tuples:

F8_IN(z0) → c2(U2'(f11_in(z0), z0))
U2'(f11_out1(z0, z1), z2) → c3(U3'(f12_in(z1), z2, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U7'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F12_IN(.(z0, z1)) → c2(U8'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
F1_IN(z0) → c7(U1'(f8_in(z0), z0))
F1_IN(z0) → c7(F8_IN(z0))
F8_IN(z0) → c7(U5'(f11_in(z0), z0))
F8_IN(z0) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2) → c7(U10'(f12_in(z1), z2, z0, z1))
U9'(f11_out1(z0, z1), z2) → c7(F12_IN(z1))

S tuples:

F8_IN(z0) → c2(U2'(f11_in(z0), z0))
U2'(f11_out1(z0, z1), z2) → c3(U3'(f12_in(z1), z2, z0, z1))
F1_IN(z0) → c7(U1'(f8_in(z0), z0))
F1_IN(z0) → c7(F8_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f8_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10

Defined Pair Symbols:

F8_IN, U2', F11_IN, U5', F12_IN, F1_IN, U9'

Compound Symbols:

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

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

Removed 7 trailing tuple parts

(27) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0, z1, z2, z3), z4) → f1_out1(z2)
f8_in(z0) → U2(f11_in(z0), z0)
f8_in(z0) → U5(f11_in(z0), z0)
f8_in(z0) → U9(f11_in(z0), z0)
U2(f11_out1(z0, z1), z2) → U3(f12_in(z1), z2, z0, z1)
U3(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2) → U6(f12_in(z1), z2, z0, z1)
U6(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U7(f12_in(z1), .(z0, z1))
f12_in(.(z0, z1)) → U8(f12_in(z1), .(z0, z1))
U7(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U8(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U9(f11_out1(z0, z1), z2) → U10(f12_in(z1), z2, z0, z1)
U10(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)

Tuples:

F8_IN(z0) → c2(U2'(f11_in(z0), z0))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
F1_IN(z0) → c7(F8_IN(z0))
F8_IN(z0) → c7(U5'(f11_in(z0), z0))
F8_IN(z0) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2) → c7(F12_IN(z1))
U2'(f11_out1(z0, z1), z2) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2) → c4
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
F12_IN(.(z0, z1)) → c2(F12_IN(z1))
F1_IN(z0) → c7
U9'(f11_out1(z0, z1), z2) → c7

S tuples:

F8_IN(z0) → c2(U2'(f11_in(z0), z0))
F1_IN(z0) → c7(F8_IN(z0))
U2'(f11_out1(z0, z1), z2) → c3
F1_IN(z0) → c7

K tuples:none
Defined Rule Symbols:

f1_in, U1, f8_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10

Defined Pair Symbols:

F8_IN, F1_IN, U9', U2', F11_IN, U5', F12_IN

Compound Symbols:

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

(28) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(z0) → c7(F8_IN(z0))
F1_IN(z0) → c7
F8_IN(z0) → c2(U2'(f11_in(z0), z0))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
F8_IN(z0) → c7(U5'(f11_in(z0), z0))
F8_IN(z0) → c7(F11_IN(z0))
U2'(f11_out1(z0, z1), z2) → c3
U9'(f11_out1(z0, z1), z2) → c7(F12_IN(z1))
U9'(f11_out1(z0, z1), z2) → c7
U5'(f11_out1(z0, z1), z2) → c4

Now S is empty

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

(30) Obligation:

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0, z1, z2, z3), z4) → f1_out1(z2)
f8_in(z0) → U2(f11_in(z0), z0)
f8_in(z0) → U5(f11_in(z0), z0)
f8_in(z0) → U9(f11_in(z0), z0)
U2(f11_out1(z0, z1), z2) → U3(f12_in(z1), z2, z0, z1)
U3(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2) → U6(f12_in(z1), z2, z0, z1)
U6(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U7(f12_in(z1), .(z0, z1))
f12_in(.(z0, z1)) → U8(f12_in(z1), .(z0, z1))
U7(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U8(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U9(f11_out1(z0, z1), z2) → U10(f12_in(z1), z2, z0, z1)
U10(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
Tuples:

F1_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F8_IN(z0) → c2(U2'(f11_in(z0), z0))
U2'(f11_out1(z0, z1), z2) → c3(U3'(f12_in(z1), z2, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F8_IN(z0) → c3(U5'(f11_in(z0), z0), F11_IN(z0))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U7'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F12_IN(.(z0, z1)) → c2(U8'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
U9'(f11_out1(z0, z1), z2) → c6(U10'(f12_in(z1), z2, z0, z1), F12_IN(z1))
S tuples:

F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F8_IN(z0) → c3(U5'(f11_in(z0), z0), F11_IN(z0))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))
K tuples:none
Defined Rule Symbols:

f1_in, U1, f8_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10

Defined Pair Symbols:

F1_IN, F8_IN, U2', F11_IN, U5', F12_IN, U9'

Compound Symbols:

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

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0, z1, z2, z3), z4) → f1_out1(z2)
f8_in(z0) → U2(f11_in(z0), z0)
f8_in(z0) → U5(f11_in(z0), z0)
f8_in(z0) → U9(f11_in(z0), z0)
U2(f11_out1(z0, z1), z2) → U3(f12_in(z1), z2, z0, z1)
U3(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2) → U6(f12_in(z1), z2, z0, z1)
U6(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U7(f12_in(z1), .(z0, z1))
f12_in(.(z0, z1)) → U8(f12_in(z1), .(z0, z1))
U7(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U8(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U9(f11_out1(z0, z1), z2) → U10(f12_in(z1), z2, z0, z1)
U10(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
Tuples:

F1_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F8_IN(z0) → c2(U2'(f11_in(z0), z0))
U2'(f11_out1(z0, z1), z2) → c3(U3'(f12_in(z1), z2, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F8_IN(z0) → c3(U5'(f11_in(z0), z0), F11_IN(z0))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U7'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F12_IN(.(z0, z1)) → c2(U8'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
U9'(f11_out1(z0, z1), z2) → c6(U10'(f12_in(z1), z2, z0, z1), F12_IN(z1))
S tuples:

F12_IN(.(z0, z1)) → c1(U7'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F12_IN(.(z0, z1)) → c2(U8'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
U9'(f11_out1(z0, z1), z2) → c6(U10'(f12_in(z1), z2, z0, z1), F12_IN(z1))
K tuples:none
Defined Rule Symbols:

f1_in, U1, f8_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10

Defined Pair Symbols:

F1_IN, F8_IN, U2', F11_IN, U5', F12_IN, U9'

Compound Symbols:

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

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

Multiplied the complexity of the problems.

(32) Complex Obligation (MULT)

(33) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0, z1, z2, z3), z4) → f1_out1(z2)
f8_in(z0) → U2(f11_in(z0), z0)
f8_in(z0) → U5(f11_in(z0), z0)
f8_in(z0) → U9(f11_in(z0), z0)
U2(f11_out1(z0, z1), z2) → U3(f12_in(z1), z2, z0, z1)
U3(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2) → U6(f12_in(z1), z2, z0, z1)
U6(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U7(f12_in(z1), .(z0, z1))
f12_in(.(z0, z1)) → U8(f12_in(z1), .(z0, z1))
U7(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U8(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U9(f11_out1(z0, z1), z2) → U10(f12_in(z1), z2, z0, z1)
U10(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)

Tuples:

F1_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F8_IN(z0) → c2(U2'(f11_in(z0), z0))
U2'(f11_out1(z0, z1), z2) → c3(U3'(f12_in(z1), z2, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F8_IN(z0) → c3(U5'(f11_in(z0), z0), F11_IN(z0))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U7'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F12_IN(.(z0, z1)) → c2(U8'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
U9'(f11_out1(z0, z1), z2) → c6(U10'(f12_in(z1), z2, z0, z1), F12_IN(z1))

S tuples:

F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F8_IN(z0) → c3(U5'(f11_in(z0), z0), F11_IN(z0))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f8_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10

Defined Pair Symbols:

F1_IN, F8_IN, U2', F11_IN, U5', F12_IN, U9'

Compound Symbols:

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

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

Split RHS of tuples not part of any SCC

(35) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0, z1, z2, z3), z4) → f1_out1(z2)
f8_in(z0) → U2(f11_in(z0), z0)
f8_in(z0) → U5(f11_in(z0), z0)
f8_in(z0) → U9(f11_in(z0), z0)
U2(f11_out1(z0, z1), z2) → U3(f12_in(z1), z2, z0, z1)
U3(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2) → U6(f12_in(z1), z2, z0, z1)
U6(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U7(f12_in(z1), .(z0, z1))
f12_in(.(z0, z1)) → U8(f12_in(z1), .(z0, z1))
U7(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U8(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U9(f11_out1(z0, z1), z2) → U10(f12_in(z1), z2, z0, z1)
U10(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)

Tuples:

F8_IN(z0) → c2(U2'(f11_in(z0), z0))
U2'(f11_out1(z0, z1), z2) → c3(U3'(f12_in(z1), z2, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U7'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F12_IN(.(z0, z1)) → c2(U8'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
F1_IN(z0) → c7(U1'(f8_in(z0), z0))
F1_IN(z0) → c7(F8_IN(z0))
F8_IN(z0) → c7(U5'(f11_in(z0), z0))
F8_IN(z0) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2) → c7(U10'(f12_in(z1), z2, z0, z1))
U9'(f11_out1(z0, z1), z2) → c7(F12_IN(z1))

S tuples:

F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))
F8_IN(z0) → c7(U5'(f11_in(z0), z0))
F8_IN(z0) → c7(F11_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f8_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10

Defined Pair Symbols:

F8_IN, U2', F11_IN, U5', F12_IN, F1_IN, U9'

Compound Symbols:

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

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

Removed 7 trailing tuple parts

(37) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0, z1, z2, z3), z4) → f1_out1(z2)
f8_in(z0) → U2(f11_in(z0), z0)
f8_in(z0) → U5(f11_in(z0), z0)
f8_in(z0) → U9(f11_in(z0), z0)
U2(f11_out1(z0, z1), z2) → U3(f12_in(z1), z2, z0, z1)
U3(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2) → U6(f12_in(z1), z2, z0, z1)
U6(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U7(f12_in(z1), .(z0, z1))
f12_in(.(z0, z1)) → U8(f12_in(z1), .(z0, z1))
U7(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U8(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U9(f11_out1(z0, z1), z2) → U10(f12_in(z1), z2, z0, z1)
U10(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)

Tuples:

F8_IN(z0) → c2(U2'(f11_in(z0), z0))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
F1_IN(z0) → c7(F8_IN(z0))
F8_IN(z0) → c7(U5'(f11_in(z0), z0))
F8_IN(z0) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2) → c7(F12_IN(z1))
U2'(f11_out1(z0, z1), z2) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2) → c4
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
F12_IN(.(z0, z1)) → c2(F12_IN(z1))
F1_IN(z0) → c7
U9'(f11_out1(z0, z1), z2) → c7

S tuples:

F8_IN(z0) → c7(U5'(f11_in(z0), z0))
F8_IN(z0) → c7(F11_IN(z0))
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2) → c4

K tuples:none
Defined Rule Symbols:

f1_in, U1, f8_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10

Defined Pair Symbols:

F8_IN, F1_IN, U9', U2', F11_IN, U5', F12_IN

Compound Symbols:

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

(38) 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.

F8_IN(z0) → c7(U5'(f11_in(z0), z0))
F8_IN(z0) → c7(F11_IN(z0))
U5'(f11_out1(z0, z1), z2) → c4

We considered the (Usable) Rules:

f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)

And the Tuples:

F8_IN(z0) → c2(U2'(f11_in(z0), z0))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
F1_IN(z0) → c7(F8_IN(z0))
F8_IN(z0) → c7(U5'(f11_in(z0), z0))
F8_IN(z0) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2) → c7(F12_IN(z1))
U2'(f11_out1(z0, z1), z2) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2) → c4
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
F12_IN(.(z0, z1)) → c2(F12_IN(z1))
F1_IN(z0) → c7
U9'(f11_out1(z0, z1), z2) → c7

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

POL(.(x₁, x₂)) = 0
POL(F11_IN(x₁)) = 0
POL(F12_IN(x₁)) = 0
POL(F1_IN(x₁)) = [3] + [2]x₁
POL(F8_IN(x₁)) = [3] + x₁
POL(U2'(x₁, x₂)) = [3]
POL(U4(x₁, x₂)) = 0
POL(U5'(x₁, x₂)) = [2]
POL(U9'(x₁, x₂)) = [1] + x₂
POL([]) = 0
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3) = 0
POL(c4) = 0
POL(c5(x₁)) = x₁
POL(c7) = 0
POL(c7(x₁)) = x₁
POL(f11_in(x₁)) = 0
POL(f11_out1(x₁, x₂)) = 0

(39) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0, z1, z2, z3), z4) → f1_out1(z2)
f8_in(z0) → U2(f11_in(z0), z0)
f8_in(z0) → U5(f11_in(z0), z0)
f8_in(z0) → U9(f11_in(z0), z0)
U2(f11_out1(z0, z1), z2) → U3(f12_in(z1), z2, z0, z1)
U3(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2) → U6(f12_in(z1), z2, z0, z1)
U6(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U7(f12_in(z1), .(z0, z1))
f12_in(.(z0, z1)) → U8(f12_in(z1), .(z0, z1))
U7(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U8(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U9(f11_out1(z0, z1), z2) → U10(f12_in(z1), z2, z0, z1)
U10(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)

Tuples:

F8_IN(z0) → c2(U2'(f11_in(z0), z0))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
F1_IN(z0) → c7(F8_IN(z0))
F8_IN(z0) → c7(U5'(f11_in(z0), z0))
F8_IN(z0) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2) → c7(F12_IN(z1))
U2'(f11_out1(z0, z1), z2) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2) → c4
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
F12_IN(.(z0, z1)) → c2(F12_IN(z1))
F1_IN(z0) → c7
U9'(f11_out1(z0, z1), z2) → c7

S tuples:

F11_IN(.(z0, z1)) → c1(F11_IN(z1))

K tuples:

F8_IN(z0) → c7(U5'(f11_in(z0), z0))
F8_IN(z0) → c7(F11_IN(z0))
U5'(f11_out1(z0, z1), z2) → c4

Defined Rule Symbols:

f1_in, U1, f8_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10

Defined Pair Symbols:

F8_IN, F1_IN, U9', U2', F11_IN, U5', F12_IN

Compound Symbols:

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

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f8_in(z0), z0)
U1(f8_out1(z0, z1, z2, z3), z4) → f1_out1(z2)
f8_in(z0) → U2(f11_in(z0), z0)
f8_in(z0) → U5(f11_in(z0), z0)
f8_in(z0) → U9(f11_in(z0), z0)
U2(f11_out1(z0, z1), z2) → U3(f12_in(z1), z2, z0, z1)
U3(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2) → U6(f12_in(z1), z2, z0, z1)
U6(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)
f12_in(z0) → f12_out1([], z0)
f12_in(.(z0, z1)) → U7(f12_in(z1), .(z0, z1))
f12_in(.(z0, z1)) → U8(f12_in(z1), .(z0, z1))
U7(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U8(f12_out1(z0, z1), .(z2, z3)) → f12_out1(.(z2, z0), z1)
U9(f11_out1(z0, z1), z2) → U10(f12_in(z1), z2, z0, z1)
U10(f12_out1(z0, z1), z2, z3, z4) → f8_out1(z3, z4, z0, z1)

Tuples:

F1_IN(z0) → c(U1'(f8_in(z0), z0), F8_IN(z0))
F8_IN(z0) → c2(U2'(f11_in(z0), z0))
U2'(f11_out1(z0, z1), z2) → c3(U3'(f12_in(z1), z2, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F8_IN(z0) → c3(U5'(f11_in(z0), z0), F11_IN(z0))
U5'(f11_out1(z0, z1), z2) → c4(U6'(f12_in(z1), z2, z0, z1))
F12_IN(.(z0, z1)) → c1(U7'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F12_IN(.(z0, z1)) → c2(U8'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
U9'(f11_out1(z0, z1), z2) → c6(U10'(f12_in(z1), z2, z0, z1), F12_IN(z1))

S tuples:

F12_IN(.(z0, z1)) → c1(U7'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F12_IN(.(z0, z1)) → c2(U8'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F8_IN(z0) → c5(U9'(f11_in(z0), z0))
U9'(f11_out1(z0, z1), z2) → c6(U10'(f12_in(z1), z2, z0, z1), F12_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f8_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10

Defined Pair Symbols:

F1_IN, F8_IN, U2', F11_IN, U5', F12_IN, U9'

Compound Symbols:

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