Runtime Complexity (innermost) proof of /tmp/tmpHCoBNg/Ex15_Luc06

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), 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 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳10 CdtProblem

↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳12 CdtProblem

↳13 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳14 CdtProblem

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

↳22 CdtProblem

↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳24 CdtProblem

↳25 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳26 CdtProblem

↳27 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳28 CdtProblem

↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 154 ms)

↳30 CdtProblem

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

↳32 CdtProblem

↳33 CdtUnreachableProof (⇔, 0 ms)

↳34 CdtProblem

↳35 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳36 CdtProblem

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

↳38 CdtProblem

↳39 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳40 CdtProblem

↳41 SIsEmptyProof (⇔, 0 ms)

↳42 BOUNDS(O(1), O(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) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

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

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

(11) 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

(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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
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, c1, c4, c4

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

Removed 1 trailing nodes:

PROPER(f(x0)) → c4

(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))
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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
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, c1, c4

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

Removed 1 trailing tuple parts

(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))
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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))

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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))

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, c1, c4, c4

(17) 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

(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(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
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, c1, c4, c4, c6, c6

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

Removed 1 trailing nodes:

PROPER(g(x0)) → c6

(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(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
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, c1, c4, c4, c6

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

Removed 1 trailing tuple parts

(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(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))

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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))

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, c1, c4, c4, c6, c6

(23) 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

(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))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
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, c1, c4, c4, c6, c6, c8, c8

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c8

(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))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
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, c1, c4, c4, c6, c6, c8

(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(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))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
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(mark(a)) → c8(TOP(ok(a)))

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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
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(mark(a)) → c8(TOP(ok(a)))

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, c1, c4, c4, c6, c6, c8, c8

(29) 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)))

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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
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(mark(a)) → c8(TOP(ok(a)))

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)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c6(x₁)) = x₁
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁)) = x₁
POL(c8(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

(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))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
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(mark(a)) → c8(TOP(ok(a)))

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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
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)))

Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, TOP, ACTIVE, PROPER

Compound Symbols:

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

(31) 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

(32) 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)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
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(mark(a)) → c8(TOP(ok(a)))
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(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
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)))

Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, ACTIVE, PROPER, TOP

Compound Symbols:

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

(33) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c1(G(mark(f(g(f(a))))))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
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)))

(34) 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(a)) → c8(TOP(ok(a)))
TOP(ok(x0)) → c9

S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))
TOP(ok(x0)) → c9

K tuples:

TOP(mark(a)) → c8(TOP(ok(a)))

Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, TOP

Compound Symbols:

c2, c3, c7, c8, c9

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

Removed 2 trailing nodes:

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

(36) 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))

S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))

K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F

Compound Symbols:

c2, c3, c7

(37) 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)) → c7(F(z0))

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

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))

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

POL(F(x₁)) = [3]x₁ + [3]x₁²
POL(G(x₁)) = 0
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [3] + x₁

(38) 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))

S tuples:

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))

K tuples:

F(ok(z0)) → c7(F(z0))

Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F

Compound Symbols:

c2, c3, c7

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

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

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))

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

G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))
F(ok(z0)) → c7(F(z0))

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

POL(F(x₁)) = x₁
POL(G(x₁)) = [4]x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(mark(x₁)) = [3] + x₁
POL(ok(x₁)) = [2] + x₁

(40) 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))

S tuples:none
K tuples:

F(ok(z0)) → c7(F(z0))
G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c3(G(z0))

Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F

Compound Symbols:

c2, c3, c7

(41) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

(33) CdtUnreachableProof (EQUIVALENT transformation)

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

(41) SIsEmptyProof (EQUIVALENT transformation)

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