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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
    ↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
    ↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 CdtProblem
    ↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳8 CdtProblem
    ↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳10 CdtProblem
    ↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳12 CdtProblem
    ↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳14 CdtProblem
    ↳15 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳16 CdtProblem
    ↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳18 CdtProblem
    ↳19 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳20 CdtProblem
    ↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 185 ms)
    ↳22 CdtProblem
    ↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳24 CdtProblem
    ↳25 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳26 CdtProblem
    ↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 766 ms)
    ↳28 CdtProblem
    ↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳30 CdtProblem
↳31 CpxTrsMatchBoundsProof (⇔, 0 ms)
↳32 BOUNDS(O(1), O(n^1))

(0) Obligation:

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

active(f(f(a))) → mark(f(g(f(a))))
active(g(X)) → g(active(X))
g(mark(X)) → mark(g(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
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(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(f(a))) → c(F(g(f(a))), G(f(a)), F(a))
ACTIVE(g(z0)) → c1(G(active(z0)), ACTIVE(z0))
G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(f(a))) → c(F(g(f(a))), G(f(a)), F(a))
ACTIVE(g(z0)) → c1(G(active(z0)), ACTIVE(z0))
G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c, c1, c2, c3, c4, c6, c7, c8, c9

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

Removed 1 trailing nodes:

ACTIVE(f(f(a))) → c(F(g(f(a))), G(f(a)), F(a))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(g(z0)) → c1(G(active(z0)), ACTIVE(z0))
G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(g(z0)) → c1(G(active(z0)), ACTIVE(z0))
G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c2, c3, c4, c6, c7, c8, c9

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

Use narrowing to replace ACTIVE(g(z0)) → c1(G(active(z0)), ACTIVE(z0)) by

ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(x0)) → c1

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(x0)) → c1
S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(x0)) → c1
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, PROPER, F, TOP, ACTIVE

Compound Symbols:

c2, c3, c4, c6, c7, c8, c9, c1, c1

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

Removed 1 trailing nodes:

ACTIVE(g(x0)) → c1

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, PROPER, F, TOP, ACTIVE

Compound Symbols:

c2, c3, c4, c6, c7, c8, c9, c1

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

Use narrowing to replace PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) by

PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(x0)) → c4

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(x0)) → c4
S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(x0)) → c4
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, PROPER, F, TOP, ACTIVE

Compound Symbols:

c2, c3, c6, c7, c8, c9, c1, c4, c4

(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

PROPER(f(x0)) → c4

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, PROPER, F, TOP, ACTIVE

Compound Symbols:

c2, c3, c6, c7, c8, c9, c1, c4

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

Use narrowing to replace PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) by

PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(x0)) → c6

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(x0)) → c6
S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(x0)) → c6
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, TOP, ACTIVE, PROPER

Compound Symbols:

c2, c3, c7, c8, c9, c1, c4, c6, c6

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

Removed 1 trailing nodes:

PROPER(g(x0)) → c6

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, TOP, ACTIVE, PROPER

Compound Symbols:

c2, c3, c7, c8, c9, c1, c4, c6

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

Use narrowing to replace TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(x0)) → c8

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(x0)) → c8
S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(x0)) → c8
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, TOP, ACTIVE, PROPER

Compound Symbols:

c2, c3, c7, c9, c1, c4, c6, c8, c8

(19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

TOP(mark(x0)) → c8

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, TOP, ACTIVE, PROPER

Compound Symbols:

c2, c3, c7, c9, c1, c4, c6, c8

(21) 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)) → c8(TOP(ok(a)), PROPER(a))
We considered the (Usable) Rules:

proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
f(ok(z0)) → ok(f(z0))
active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
And the Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0   
POL(F(x1)) = 0   
POL(G(x1)) = 0   
POL(PROPER(x1)) = 0   
POL(TOP(x1)) = [2]x1   
POL(a) = [1]   
POL(active(x1)) = 0   
POL(c1(x1, x2)) = x1 + x2   
POL(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(x1, x2)) = x1 + x2   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1)) = x1   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1)) = 0   
POL(g(x1)) = 0   
POL(mark(x1)) = x1   
POL(ok(x1)) = 0   
POL(proper(x1)) = 0   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
K tuples:

TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, TOP, ACTIVE, PROPER

Compound Symbols:

c2, c3, c7, c9, c1, c4, c6, c8

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

Use narrowing to replace TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(g(z0))) → c9(TOP(g(active(z0))), ACTIVE(g(z0)))
TOP(ok(x0)) → c9

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(g(z0))) → c9(TOP(g(active(z0))), ACTIVE(g(z0)))
TOP(ok(x0)) → c9
S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(g(z0))) → c9(TOP(g(active(z0))), ACTIVE(g(z0)))
TOP(ok(x0)) → c9
K tuples:

TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, ACTIVE, PROPER, TOP

Compound Symbols:

c2, c3, c7, c1, c4, c6, c8, c9, c9

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

Removed 2 trailing nodes:

TOP(ok(x0)) → c9
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(g(z0))) → c9(TOP(g(active(z0))), ACTIVE(g(z0)))
S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(g(z0))) → c9(TOP(g(active(z0))), ACTIVE(g(z0)))
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, ACTIVE, PROPER, TOP

Compound Symbols:

c2, c3, c7, c1, c4, c6, c8, c9

(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.

TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
We considered the (Usable) Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
And the Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(g(z0))) → c9(TOP(g(active(z0))), ACTIVE(g(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0   
POL(F(x1)) = 0   
POL(G(x1)) = 0   
POL(PROPER(x1)) = 0   
POL(TOP(x1)) = [4]x1   
POL(a) = [1]   
POL(active(x1)) = 0   
POL(c1(x1, x2)) = x1 + x2   
POL(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(x1, x2)) = x1 + x2   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1)) = x1   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1)) = [4]x1   
POL(g(x1)) = 0   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = x1   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(g(z0))) → c9(TOP(g(active(z0))), ACTIVE(g(z0)))
S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(g(z0))) → c9(TOP(g(active(z0))), ACTIVE(g(z0)))
K tuples:

TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, ACTIVE, PROPER, TOP

Compound Symbols:

c2, c3, c7, c1, c4, c6, c8, c9

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

Use narrowing to replace ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))), ACTIVE(f(f(a)))) by

ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(g(z0))) → c9(TOP(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(g(z0))) → c9(TOP(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
K tuples:

TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, ACTIVE, PROPER, TOP

Compound Symbols:

c2, c3, c7, c1, c4, c6, c8, c9, c1

(31) CpxTrsMatchBoundsProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2.
The certificate found is represented by the following graph.
Start state: 1567
Accept states: [1568, 1569, 1570, 1571, 1572]
Transitions:
1567→1568[active_1|0]
1567→1569[g_1|0]
1567→1570[proper_1|0]
1567→1571[f_1|0]
1567→1572[top_1|0]
1567→1567[a|0, mark_1|0, ok_1|0]
1567→1573[a|1]
1567→1574[g_1|1]
1567→1575[proper_1|1]
1567→1576[g_1|1]
1567→1577[f_1|1]
1567→1578[active_1|1]
1573→1570[ok_1|1]
1573→1575[ok_1|1]
1573→1579[active_1|2]
1574→1569[mark_1|1]
1574→1574[mark_1|1]
1574→1576[mark_1|1]
1575→1572[top_1|1]
1576→1569[ok_1|1]
1576→1574[ok_1|1]
1576→1576[ok_1|1]
1577→1571[ok_1|1]
1577→1577[ok_1|1]
1578→1572[top_1|1]
1579→1572[top_1|2]

(32) BOUNDS(O(1), O(n^1))