The Runtime Complexity (innermost) of the given CpxTRS could be proven to be 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))))
mark(f(X)) → active(f(X))
mark(a) → active(a)
mark(g(X)) → active(g(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(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))))
mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
MARK(a) → c2(ACTIVE(a))
MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))

S tuples:

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
MARK(a) → c2(ACTIVE(a))
MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))

K tuples:none
Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

ACTIVE, MARK, F, G

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7

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

Removed 1 trailing nodes:

MARK(a) → c2(ACTIVE(a))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))

S tuples:

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))

K tuples:none
Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

ACTIVE, MARK, F, G

Compound Symbols:

c, c1, c3, c4, c5, c6, c7

(5) 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.

MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))

We considered the (Usable) Rules:

mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
active(f(f(a))) → mark(f(g(f(a))))
g(active(z0)) → g(z0)
g(mark(z0)) → g(z0)

And the Tuples:

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))

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

POL(ACTIVE(x₁)) = [2]   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(MARK(x₁)) = [2] + [4]x₁   
POL(a) = 0   
POL(active(x₁)) = [2] + [3]x₁   
POL(c(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(f(x₁)) = 0   
POL(g(x₁)) = [2] + [4]x₁   
POL(mark(x₁)) = [4] + [4]x₁   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))

S tuples:

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))

K tuples:

MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

ACTIVE, MARK, F, G

Compound Symbols:

c, c1, c3, c4, c5, c6, c7

(7) 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)) → c6(G(z0))
G(active(z0)) → c7(G(z0))

We considered the (Usable) Rules:

mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
active(f(f(a))) → mark(f(g(f(a))))
g(active(z0)) → g(z0)
g(mark(z0)) → g(z0)

And the Tuples:

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(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)) = [2]x₁   
POL(MARK(x₁)) = [4]x₁   
POL(a) = 0   
POL(active(x₁)) = [1] + x₁   
POL(c(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(f(x₁)) = 0   
POL(g(x₁)) = [4] + [4]x₁   
POL(mark(x₁)) = [1] + [2]x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))

S tuples:

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))

K tuples:

MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

ACTIVE, MARK, F, G

Compound Symbols:

c, c1, c3, c4, c5, c6, c7

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

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

F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))

We considered the (Usable) Rules:

mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
active(f(f(a))) → mark(f(g(f(a))))
g(active(z0)) → g(z0)
g(mark(z0)) → g(z0)

And the Tuples:

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))

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

POL(ACTIVE(x₁)) = 0   
POL(F(x₁)) = [2]x₁   
POL(G(x₁)) = 0   
POL(MARK(x₁)) = x₁   
POL(a) = 0   
POL(active(x₁)) = [4] + [4]x₁   
POL(c(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(f(x₁)) = [2]x₁   
POL(g(x₁)) = x₁   
POL(mark(x₁)) = [2] + [2]x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))

S tuples:

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))

K tuples:

MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

ACTIVE, MARK, F, G

Compound Symbols:

c, c1, c3, c4, c5, c6, c7

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

Use narrowing to replace ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a)) by

ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))

S tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))

K tuples:

MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

MARK, F, G, ACTIVE

Compound Symbols:

c1, c3, c4, c5, c6, c7, c

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

Use narrowing to replace MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0)) by

MARK(g(z0)) → c3(ACTIVE(g(z0)), G(mark(z0)), MARK(z0))
MARK(g(f(z0))) → c3(ACTIVE(g(active(f(z0)))), G(mark(f(z0))), MARK(f(z0)))
MARK(g(a)) → c3(ACTIVE(g(active(a))), G(mark(a)), MARK(a))
MARK(g(g(z0))) → c3(ACTIVE(g(active(g(mark(z0))))), G(mark(g(z0))), MARK(g(z0)))
MARK(g(x0)) → c3

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))
MARK(g(z0)) → c3(ACTIVE(g(z0)), G(mark(z0)), MARK(z0))
MARK(g(f(z0))) → c3(ACTIVE(g(active(f(z0)))), G(mark(f(z0))), MARK(f(z0)))
MARK(g(a)) → c3(ACTIVE(g(active(a))), G(mark(a)), MARK(a))
MARK(g(g(z0))) → c3(ACTIVE(g(active(g(mark(z0))))), G(mark(g(z0))), MARK(g(z0)))
MARK(g(x0)) → c3

S tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))

K tuples:

MARK(g(z0)) → c3(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

MARK, F, G, ACTIVE

Compound Symbols:

c1, c4, c5, c6, c7, c, c3, c3

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

Removed 1 trailing nodes:

MARK(g(x0)) → c3

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))
MARK(g(z0)) → c3(ACTIVE(g(z0)), G(mark(z0)), MARK(z0))
MARK(g(f(z0))) → c3(ACTIVE(g(active(f(z0)))), G(mark(f(z0))), MARK(f(z0)))
MARK(g(a)) → c3(ACTIVE(g(active(a))), G(mark(a)), MARK(a))
MARK(g(g(z0))) → c3(ACTIVE(g(active(g(mark(z0))))), G(mark(g(z0))), MARK(g(z0)))

S tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))

K tuples:

G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

MARK, F, G, ACTIVE

Compound Symbols:

c1, c4, c5, c6, c7, c, c3

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

Use narrowing to replace MARK(g(z0)) → c3(ACTIVE(g(z0)), G(mark(z0)), MARK(z0)) by

MARK(g(x0)) → c3(G(mark(x0)), MARK(x0))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))
MARK(g(f(z0))) → c3(ACTIVE(g(active(f(z0)))), G(mark(f(z0))), MARK(f(z0)))
MARK(g(a)) → c3(ACTIVE(g(active(a))), G(mark(a)), MARK(a))
MARK(g(g(z0))) → c3(ACTIVE(g(active(g(mark(z0))))), G(mark(g(z0))), MARK(g(z0)))
MARK(g(x0)) → c3(G(mark(x0)), MARK(x0))

S tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))

K tuples:

G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

MARK, F, G, ACTIVE

Compound Symbols:

c1, c4, c5, c6, c7, c, c3, c3

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

Use narrowing to replace MARK(g(f(z0))) → c3(ACTIVE(g(active(f(z0)))), G(mark(f(z0))), MARK(f(z0))) by

MARK(g(f(x0))) → c3(ACTIVE(g(f(x0))), G(mark(f(x0))), MARK(f(x0)))
MARK(g(f(f(a)))) → c3(ACTIVE(g(mark(f(g(f(a)))))), G(mark(f(f(a)))), MARK(f(f(a))))
MARK(g(f(x0))) → c3(G(mark(f(x0))), MARK(f(x0)))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))
MARK(g(a)) → c3(ACTIVE(g(active(a))), G(mark(a)), MARK(a))
MARK(g(g(z0))) → c3(ACTIVE(g(active(g(mark(z0))))), G(mark(g(z0))), MARK(g(z0)))
MARK(g(x0)) → c3(G(mark(x0)), MARK(x0))
MARK(g(f(x0))) → c3(ACTIVE(g(f(x0))), G(mark(f(x0))), MARK(f(x0)))
MARK(g(f(f(a)))) → c3(ACTIVE(g(mark(f(g(f(a)))))), G(mark(f(f(a)))), MARK(f(f(a))))
MARK(g(f(x0))) → c3(G(mark(f(x0))), MARK(f(x0)))

S tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))

K tuples:

G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

MARK, F, G, ACTIVE

Compound Symbols:

c1, c4, c5, c6, c7, c, c3, c3

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

Use narrowing to replace MARK(g(a)) → c3(ACTIVE(g(active(a))), G(mark(a)), MARK(a)) by

MARK(g(a)) → c3(ACTIVE(g(a)), G(mark(a)), MARK(a))
MARK(g(a)) → c3(G(mark(a)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))
MARK(g(g(z0))) → c3(ACTIVE(g(active(g(mark(z0))))), G(mark(g(z0))), MARK(g(z0)))
MARK(g(x0)) → c3(G(mark(x0)), MARK(x0))
MARK(g(f(x0))) → c3(ACTIVE(g(f(x0))), G(mark(f(x0))), MARK(f(x0)))
MARK(g(f(f(a)))) → c3(ACTIVE(g(mark(f(g(f(a)))))), G(mark(f(f(a)))), MARK(f(f(a))))
MARK(g(f(x0))) → c3(G(mark(f(x0))), MARK(f(x0)))
MARK(g(a)) → c3(ACTIVE(g(a)), G(mark(a)), MARK(a))
MARK(g(a)) → c3(G(mark(a)))

S tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))

K tuples:

G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

MARK, F, G, ACTIVE

Compound Symbols:

c1, c4, c5, c6, c7, c, c3, c3, c3

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

Use narrowing to replace MARK(g(g(z0))) → c3(ACTIVE(g(active(g(mark(z0))))), G(mark(g(z0))), MARK(g(z0))) by

MARK(g(g(x0))) → c3(ACTIVE(g(g(mark(x0)))), G(mark(g(x0))), MARK(g(x0)))
MARK(g(g(z0))) → c3(ACTIVE(g(active(g(z0)))), G(mark(g(z0))), MARK(g(z0)))
MARK(g(g(f(z0)))) → c3(ACTIVE(g(active(g(active(f(z0)))))), G(mark(g(f(z0)))), MARK(g(f(z0))))
MARK(g(g(a))) → c3(ACTIVE(g(active(g(active(a))))), G(mark(g(a))), MARK(g(a)))
MARK(g(g(g(z0)))) → c3(ACTIVE(g(active(g(active(g(mark(z0))))))), G(mark(g(g(z0)))), MARK(g(g(z0))))
MARK(g(g(x0))) → c3(G(mark(g(x0))))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))
MARK(g(x0)) → c3(G(mark(x0)), MARK(x0))
MARK(g(f(x0))) → c3(ACTIVE(g(f(x0))), G(mark(f(x0))), MARK(f(x0)))
MARK(g(f(f(a)))) → c3(ACTIVE(g(mark(f(g(f(a)))))), G(mark(f(f(a)))), MARK(f(f(a))))
MARK(g(f(x0))) → c3(G(mark(f(x0))), MARK(f(x0)))
MARK(g(a)) → c3(ACTIVE(g(a)), G(mark(a)), MARK(a))
MARK(g(a)) → c3(G(mark(a)))
MARK(g(g(x0))) → c3(ACTIVE(g(g(mark(x0)))), G(mark(g(x0))), MARK(g(x0)))
MARK(g(g(z0))) → c3(ACTIVE(g(active(g(z0)))), G(mark(g(z0))), MARK(g(z0)))
MARK(g(g(f(z0)))) → c3(ACTIVE(g(active(g(active(f(z0)))))), G(mark(g(f(z0)))), MARK(g(f(z0))))
MARK(g(g(a))) → c3(ACTIVE(g(active(g(active(a))))), G(mark(g(a))), MARK(g(a)))
MARK(g(g(g(z0)))) → c3(ACTIVE(g(active(g(active(g(mark(z0))))))), G(mark(g(g(z0)))), MARK(g(g(z0))))
MARK(g(g(x0))) → c3(G(mark(g(x0))))

S tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))

K tuples:

G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

MARK, F, G, ACTIVE

Compound Symbols:

c1, c4, c5, c6, c7, c, c3, c3, c3

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

Used rewriting to replace MARK(g(f(x0))) → c3(ACTIVE(g(f(x0))), G(mark(f(x0))), MARK(f(x0))) by MARK(g(f(z0))) → c3(ACTIVE(g(f(z0))), G(active(f(z0))), MARK(f(z0)))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(z0))
mark(a) → active(a)
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))
MARK(g(x0)) → c3(G(mark(x0)), MARK(x0))
MARK(g(f(f(a)))) → c3(ACTIVE(g(mark(f(g(f(a)))))), G(mark(f(f(a)))), MARK(f(f(a))))
MARK(g(f(x0))) → c3(G(mark(f(x0))), MARK(f(x0)))
MARK(g(a)) → c3(ACTIVE(g(a)), G(mark(a)), MARK(a))
MARK(g(a)) → c3(G(mark(a)))
MARK(g(g(x0))) → c3(ACTIVE(g(g(mark(x0)))), G(mark(g(x0))), MARK(g(x0)))
MARK(g(g(z0))) → c3(ACTIVE(g(active(g(z0)))), G(mark(g(z0))), MARK(g(z0)))
MARK(g(g(f(z0)))) → c3(ACTIVE(g(active(g(active(f(z0)))))), G(mark(g(f(z0)))), MARK(g(f(z0))))
MARK(g(g(a))) → c3(ACTIVE(g(active(g(active(a))))), G(mark(g(a))), MARK(g(a)))
MARK(g(g(g(z0)))) → c3(ACTIVE(g(active(g(active(g(mark(z0))))))), G(mark(g(g(z0)))), MARK(g(g(z0))))
MARK(g(g(x0))) → c3(G(mark(g(x0))))
MARK(g(f(z0))) → c3(ACTIVE(g(f(z0))), G(active(f(z0))), MARK(f(z0)))

S tuples:

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(z0))
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))))

K tuples:

G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

MARK, F, G, ACTIVE

Compound Symbols:

c1, c4, c5, c6, c7, c, c3, c3, c3

(27) 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 1.
The certificate found is represented by the following graph.

Start state: 1521
Accept states: [1522, 1523, 1524, 1525]
Transitions:
1521→1522[active_1|0]
1521→1523[mark_1|0]
1521→1524[f_1|0]
1521→1525[g_1|0]
1521→1521[a|0]
1521→1526[a|1]
1526→1523[active_1|1]

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