Runtime Complexity (innermost) proof of /scratch/tmpbcxCXs/Ex15_Luc06

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))

↳2 CdtProblem

↳3 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳4 CdtProblem

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

↳6 CdtProblem

↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

↳8 CdtProblem

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

↳10 CdtProblem

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

↳12 CdtProblem

↳13 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳14 CdtProblem

↳15 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

↳16 CdtProblem

↳17 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))

↳18 CdtProblem

↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

↳20 CdtProblem

↳21 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳22 CdtProblem

↳23 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

↳24 CdtProblem

↳25 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))

↳26 CdtProblem

↳27 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

↳28 CdtProblem

↳29 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳30 CdtProblem

↳31 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))

↳32 CdtProblem

↳33 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

↳34 CdtProblem

↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

↳36 CdtProblem

↳37 CdtUnreachableProof (⇔)

↳38 CdtProblem

↳39 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳40 CdtProblem

↳41 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

↳42 CdtProblem

↳43 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

↳44 CdtProblem

↳45 SIsEmptyProof (⇔)

↳46 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) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 9 dangling 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)))

(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)))

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

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

Split RHS of tuples not part of any SCC

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

(9) CdtGraphRemoveTrailingProof (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)))) → c(G(mark(f(g(f(a))))))
ACTIVE(g(f(f(a)))) → c

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

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, c, c

(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)))

(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)))) → c(G(mark(f(g(f(a))))))
ACTIVE(g(f(f(a)))) → c
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)))) → c(G(mark(f(g(f(a))))))
ACTIVE(g(f(f(a)))) → c
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, c, c, c4

(13) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 12 dangling nodes:

ACTIVE(g(f(f(a)))) → c

(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)))) → c(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)))) → c(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, c, c4

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

Split RHS of tuples not part of any SCC

(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)))) → c(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)) → c5(F(ok(a)))
PROPER(f(a)) → c5(PROPER(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)))) → c(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)) → c5(F(ok(a)))
PROPER(f(a)) → c5(PROPER(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, c, c4, c5

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

Removed 1 trailing tuple parts

(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))
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)))) → c(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)) → c5(F(ok(a)))
PROPER(f(a)) → c5

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)))) → c(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)) → c5(F(ok(a)))
PROPER(f(a)) → c5

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, c, c4, c5, c5

(19) 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)))

(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)))) → c(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)) → c5(F(ok(a)))
PROPER(f(a)) → c5
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)))) → c(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)) → c5(F(ok(a)))
PROPER(f(a)) → c5
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, c, c4, c5, c5, c6

(21) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 14 dangling nodes:

PROPER(f(a)) → c5

(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)))) → c(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)) → c5(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)))) → c(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)) → c5(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, c, c4, c5, c6

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

Split RHS of tuples not part of any SCC

(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(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)))) → c(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)) → c5(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)) → c10(G(ok(a)))
PROPER(g(a)) → c10(PROPER(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)))) → c(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)) → c5(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)) → c10(G(ok(a)))
PROPER(g(a)) → c10(PROPER(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, c, c4, c5, c6, c10

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

Removed 1 trailing tuple parts

(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(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)))) → c(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)) → c5(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)) → c10(G(ok(a)))
PROPER(g(a)) → c10

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)))) → c(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)) → c5(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)) → c10(G(ok(a)))
PROPER(g(a)) → c10

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, c, c4, c5, c6, c10, c10

(27) 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)))

(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)))) → c(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)) → c5(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)) → c10(G(ok(a)))
PROPER(g(a)) → c10
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)))) → c(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)) → c5(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)) → c10(G(ok(a)))
PROPER(g(a)) → c10
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, c, c4, c5, c6, c10, c10, c8

(29) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 16 dangling nodes:

PROPER(g(a)) → c10

(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)))) → c(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)) → c5(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)) → c10(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)))) → c(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)) → c5(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)) → c10(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, c, c4, c5, c6, c10, c8

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

Removed 1 trailing tuple parts

(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))
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)))) → c(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)) → c5(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)) → c10(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)))) → c(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)) → c5(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)) → c10(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, c, c4, c5, c6, c10, c8, c8

(33) 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)))) → c(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)) → c5(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)) → c10(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(c(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(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

(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(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)))) → c(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)) → c5(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)) → c10(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)))) → c(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)) → c5(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)) → c10(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, c, c4, c5, c6, c10, c8, c8

(35) 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)))

(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))
ACTIVE(g(g(z0))) → c1(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(f(a)))) → c(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)) → c5(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)) → c10(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)))

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)))) → c(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)) → c5(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)) → c10(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)))

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, c, c4, c5, c6, c10, c8, c8, c9

(37) 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)))) → c(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)) → c5(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)) → c10(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)))

(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))
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))

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

(39) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 4 dangling nodes:

TOP(mark(a)) → c8(TOP(ok(a)))

(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:

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

(41) 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(ok(z0)) → c3(G(z0))
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)) = [4]x₁
POL(G(x₁)) = [4]x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = [1] + x₁

(42) 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))

K tuples:

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

Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F

Compound Symbols:

c2, c3, c7

(43) 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))

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)) = [5]x₁
POL(G(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(44) 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:

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

Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F

Compound Symbols:

c2, c3, c7

(45) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

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

(12) Obligation:

(13) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

(21) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

(29) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

(37) CdtUnreachableProof (EQUIVALENT transformation)

(38) Obligation:

(39) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(40) Obligation:

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

(42) Obligation:

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

(44) Obligation:

(45) SIsEmptyProof (EQUIVALENT transformation)

(46) BOUNDS(O(1), O(1))