The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳4 CdtProblem
↳5 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳6 CdtProblem
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳8 CdtProblem
↳9 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳10 CdtProblem
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳12 CdtProblem
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳14 CdtProblem
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳16 CdtProblem
↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳18 CdtProblem
↳19 CdtUnreachableProof (⇔)
↳20 CdtProblem
↳21 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳22 CdtProblem
↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳24 CdtProblem
↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳26 CdtProblem
↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳28 CdtProblem
↳29 SIsEmptyProof (⇔)
↳30 BOUNDS(O(1), O(1))

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(active(X1), X2, X3)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(mark(X1), X2, X3) → mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
PROPER(f(z0, z1, z2)) → c9(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
PROPER(f(z0, z1, z2)) → c9(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9, c13, c14

(3) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(f(z0, z1, z2)) → c9(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c13, c14, c9

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

Removed 9 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c13, c14, c9, c9

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

Use narrowing to replace TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(a)) → c13(TOP(ok(a)), PROPER(a))
TOP(mark(b)) → c13(TOP(ok(b)), PROPER(b))
TOP(mark(c)) → c13(TOP(ok(c)), PROPER(c))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(a)) → c13(TOP(ok(a)), PROPER(a))
TOP(mark(b)) → c13(TOP(ok(b)), PROPER(b))
TOP(mark(c)) → c13(TOP(ok(c)), PROPER(c))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(a)) → c13(TOP(ok(a)), PROPER(a))
TOP(mark(b)) → c13(TOP(ok(b)), PROPER(b))
TOP(mark(c)) → c13(TOP(ok(c)), PROPER(c))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c14, c9, c9, c13

(9) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(b)) → c13(TOP(ok(b)))
TOP(mark(c)) → c13(TOP(ok(c)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(b)) → c13(TOP(ok(b)))
TOP(mark(c)) → c13(TOP(ok(c)))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c14, c9, c9, c13, c13

(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.

TOP(mark(a)) → c13(TOP(ok(a)))
We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
And the Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(b)) → c13(TOP(ok(b)))
TOP(mark(c)) → c13(TOP(ok(c)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0   
POL(F(x1, x2, x3)) = 0   
POL(PROPER(x1)) = 0   
POL(TOP(x1)) = x1   
POL(a) = 0   
POL(active(x1)) = x1   
POL(b) = [4]   
POL(c) = [4]   
POL(c1(x1)) = x1   
POL(c13(x1)) = x1   
POL(c13(x1, x2)) = x1 + x2   
POL(c14(x1, x2)) = x1 + x2   
POL(c4(x1, x2)) = x1 + x2   
POL(c5(x1, x2)) = x1 + x2   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1, x2, x3)) = x1 + x2 + x3   
POL(c9(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(f(x1, x2, x3)) = [4]   
POL(mark(x1)) = [4]   
POL(ok(x1)) = x1   
POL(proper(x1)) = [4]   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(b)) → c13(TOP(ok(b)))
TOP(mark(c)) → c13(TOP(ok(c)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(b)) → c13(TOP(ok(b)))
TOP(mark(c)) → c13(TOP(ok(c)))
K tuples:

TOP(mark(a)) → c13(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c14, c9, c9, c13, c13

(13) 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.

TOP(mark(c)) → c13(TOP(ok(c)))
We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
And the Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(b)) → c13(TOP(ok(b)))
TOP(mark(c)) → c13(TOP(ok(c)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0   
POL(F(x1, x2, x3)) = 0   
POL(PROPER(x1)) = 0   
POL(TOP(x1)) = x1   
POL(a) = 0   
POL(active(x1)) = 0   
POL(b) = 0   
POL(c) = [1]   
POL(c1(x1)) = x1   
POL(c13(x1)) = x1   
POL(c13(x1, x2)) = x1 + x2   
POL(c14(x1, x2)) = x1 + x2   
POL(c4(x1, x2)) = x1 + x2   
POL(c5(x1, x2)) = x1 + x2   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1, x2, x3)) = x1 + x2 + x3   
POL(c9(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(f(x1, x2, x3)) = 0   
POL(mark(x1)) = x1   
POL(ok(x1)) = 0   
POL(proper(x1)) = [4]   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(b)) → c13(TOP(ok(b)))
TOP(mark(c)) → c13(TOP(ok(c)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(b)) → c13(TOP(ok(b)))
K tuples:

TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(c)) → c13(TOP(ok(c)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c14, c9, c9, c13, c13

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

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

TOP(mark(b)) → c13(TOP(ok(b)))
We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
And the Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(b)) → c13(TOP(ok(b)))
TOP(mark(c)) → c13(TOP(ok(c)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0   
POL(F(x1, x2, x3)) = 0   
POL(PROPER(x1)) = 0   
POL(TOP(x1)) = x1   
POL(a) = 0   
POL(active(x1)) = x1   
POL(b) = 0   
POL(c) = [1]   
POL(c1(x1)) = x1   
POL(c13(x1)) = x1   
POL(c13(x1, x2)) = x1 + x2   
POL(c14(x1, x2)) = x1 + x2   
POL(c4(x1, x2)) = x1 + x2   
POL(c5(x1, x2)) = x1 + x2   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1, x2, x3)) = x1 + x2 + x3   
POL(c9(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(f(x1, x2, x3)) = [1]   
POL(mark(x1)) = [1]   
POL(ok(x1)) = x1   
POL(proper(x1)) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(b)) → c13(TOP(ok(b)))
TOP(mark(c)) → c13(TOP(ok(c)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
K tuples:

TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(c)) → c13(TOP(ok(c)))
TOP(mark(b)) → c13(TOP(ok(b)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c14, c9, c9, c13, c13

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(a, b, z0))) → c14(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0)))
TOP(ok(c)) → c14(TOP(mark(a)), ACTIVE(c))
TOP(ok(c)) → c14(TOP(mark(b)), ACTIVE(c))
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(active(z0), z1, z2)), ACTIVE(f(z0, z1, z2)))
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(b)) → c13(TOP(ok(b)))
TOP(mark(c)) → c13(TOP(ok(c)))
TOP(ok(f(a, b, z0))) → c14(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0)))
TOP(ok(c)) → c14(TOP(mark(a)), ACTIVE(c))
TOP(ok(c)) → c14(TOP(mark(b)), ACTIVE(c))
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(active(z0), z1, z2)), ACTIVE(f(z0, z1, z2)))
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(ok(f(a, b, z0))) → c14(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0)))
TOP(ok(c)) → c14(TOP(mark(a)), ACTIVE(c))
TOP(ok(c)) → c14(TOP(mark(b)), ACTIVE(c))
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(active(z0), z1, z2)), ACTIVE(f(z0, z1, z2)))
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
K tuples:

TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(c)) → c13(TOP(ok(c)))
TOP(mark(b)) → c13(TOP(ok(b)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9, c9, c13, c13, c14

(19) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2))
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(ok(f(a, b, z0))) → c14(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0)))
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(active(z0), z1, z2)), ACTIVE(f(z0, z1, z2)))
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(b)) → c13(TOP(ok(b)))
TOP(mark(c)) → c13(TOP(ok(c)))
TOP(ok(c)) → c14(TOP(mark(a)), ACTIVE(c))
TOP(ok(c)) → c14(TOP(mark(b)), ACTIVE(c))
S tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(c)) → c14(TOP(mark(a)), ACTIVE(c))
TOP(ok(c)) → c14(TOP(mark(b)), ACTIVE(c))
K tuples:

TOP(mark(a)) → c13(TOP(ok(a)))
TOP(mark(c)) → c13(TOP(ok(c)))
TOP(mark(b)) → c13(TOP(ok(b)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP

Compound Symbols:

c6, c7, c8, c13, c14

(21) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 5 of 8 dangling nodes:

TOP(ok(c)) → c14(TOP(mark(b)), ACTIVE(c))
TOP(ok(c)) → c14(TOP(mark(a)), ACTIVE(c))
TOP(mark(b)) → c13(TOP(ok(b)))
TOP(mark(c)) → c13(TOP(ok(c)))
TOP(mark(a)) → c13(TOP(ok(a)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
S tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c6, c7, c8

(23) 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.

F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2, x3)) = [3]x2 + [3]x22   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(mark(x1)) = 0   
POL(ok(x1)) = [3] + x1   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
S tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
K tuples:

F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c6, c7, c8

(25) 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.

F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2, x3)) = x1 + [3]x2   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(mark(x1)) = [4] + x1   
POL(ok(x1)) = x1   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
S tuples:

F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
K tuples:

F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c6, c7, c8

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

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

F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2, x3)) = [4]x2 + x3   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = x1   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0))
active(c) → mark(a)
active(c) → mark(b)
active(f(z0, z1, z2)) → f(active(z0), z1, z2)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(mark(z0), z1, z2) → mark(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
S tuples:none
K tuples:

F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c6, c7, c8

(29) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(30) BOUNDS(O(1), O(1))