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(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

S tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

K tuples:none
Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

ACTIVE, MARK, F, G, H

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

(3) 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(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

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

POL(ACTIVE(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = 0   
POL(MARK(x₁)) = x₁   
POL(active(x₁)) = 0   
POL(c(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c1(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c2(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(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f(x₁)) = [4] + x₁   
POL(g(x₁)) = 0   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = 0   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

S tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

K tuples:

MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

ACTIVE, MARK, F, G, H

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

(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(h(z0)) → c3(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

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

POL(ACTIVE(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = 0   
POL(MARK(x₁)) = [2]x₁   
POL(active(x₁)) = [4] + [4]x₁   
POL(c(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c1(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c2(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(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f(x₁)) = [2] + x₁   
POL(g(x₁)) = 0   
POL(h(x₁)) = [1] + [4]x₁   
POL(mark(x₁)) = 0   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

S tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

K tuples:

MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

ACTIVE, MARK, F, G, H

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

(7) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

S tuples:

F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

K tuples:

MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

ACTIVE, MARK, F, G, H

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

(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))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

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

POL(ACTIVE(x₁)) = [4] + x₁   
POL(F(x₁)) = [4] + [2]x₁   
POL(G(x₁)) = 0   
POL(H(x₁)) = 0   
POL(MARK(x₁)) = [4] + [5]x₁   
POL(active(x₁)) = x₁   
POL(c(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c1(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c2(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(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f(x₁)) = [4] + [3]x₁   
POL(g(x₁)) = 0   
POL(h(x₁)) = [4] + [4]x₁   
POL(mark(x₁)) = [1] + [2]x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

S tuples:

F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

K tuples:

MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c4(F(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

ACTIVE, MARK, F, G, H

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

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

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

H(mark(z0)) → c8(H(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

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

POL(ACTIVE(x₁)) = x₁   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = x₁   
POL(MARK(x₁)) = [3]x₁   
POL(active(x₁)) = x₁   
POL(c(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c1(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c2(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(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f(x₁)) = [3] + [4]x₁   
POL(g(x₁)) = 0   
POL(h(x₁)) = [4] + [5]x₁   
POL(mark(x₁)) = [1] + [2]x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

S tuples:

F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
H(active(z0)) → c9(H(z0))

K tuples:

MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c4(F(z0))
H(mark(z0)) → c8(H(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

ACTIVE, MARK, F, G, H

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

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

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

H(active(z0)) → c9(H(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

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

POL(ACTIVE(x₁)) = [1] + x₁   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = x₁   
POL(MARK(x₁)) = [1] + [3]x₁   
POL(active(x₁)) = [1] + x₁   
POL(c(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c1(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c2(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(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f(x₁)) = [4] + [4]x₁   
POL(g(x₁)) = 0   
POL(h(x₁)) = [5] + [5]x₁   
POL(mark(x₁)) = [1] + [2]x₁   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

S tuples:

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

K tuples:

MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c4(F(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

ACTIVE, MARK, F, G, H

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

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

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

F(active(z0)) → c5(F(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

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

POL(ACTIVE(x₁)) = [3] + x₁   
POL(F(x₁)) = x₁   
POL(G(x₁)) = 0   
POL(H(x₁)) = 0   
POL(MARK(x₁)) = [5]x₁   
POL(active(x₁)) = [2] + x₁   
POL(c(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c1(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c2(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(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f(x₁)) = [4] + [4]x₁   
POL(g(x₁)) = [1]   
POL(h(x₁)) = [4] + [4]x₁   
POL(mark(x₁)) = [1] + [3]x₁   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))

S tuples:

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

K tuples:

MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c4(F(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

ACTIVE, MARK, F, G, H

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

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

Use narrowing to replace ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0)) by

ACTIVE(f(x0)) → c(MARK(g(h(f(x0)))), F(x0))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
ACTIVE(f(x0)) → c(MARK(g(h(f(x0)))), F(x0))

S tuples:

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

K tuples:

MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c4(F(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

MARK, F, G, H, ACTIVE

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c

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

Use narrowing to replace MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0)) by

MARK(f(z0)) → c1(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c1(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(g(z0))) → c1(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(h(z0))) → c1(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))
MARK(f(x0)) → c1

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
ACTIVE(f(x0)) → c(MARK(g(h(f(x0)))), F(x0))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c1(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(g(z0))) → c1(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(h(z0))) → c1(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))
MARK(f(x0)) → c1

S tuples:

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

K tuples:

MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(z0)) → c(MARK(g(h(f(z0)))), G(h(f(z0))), H(f(z0)), F(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c4(F(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

MARK, F, G, H, ACTIVE

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c, c1, c1

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

Removed 1 trailing nodes:

MARK(f(x0)) → c1

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
ACTIVE(f(x0)) → c(MARK(g(h(f(x0)))), F(x0))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c1(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(g(z0))) → c1(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(h(z0))) → c1(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))

S tuples:

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

K tuples:

MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c4(F(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

MARK, F, G, H, ACTIVE

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c, c1

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

Use narrowing to replace MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0)) by

MARK(g(x0)) → c2(G(x0))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(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))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
ACTIVE(f(x0)) → c(MARK(g(h(f(x0)))), F(x0))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c1(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(g(z0))) → c1(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(h(z0))) → c1(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))
MARK(g(x0)) → c2(G(x0))

S tuples:

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

K tuples:

MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c4(F(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

MARK, F, G, H, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c, c1, c2

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

Use narrowing to replace MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0)) by

MARK(h(z0)) → c3(ACTIVE(h(z0)), H(mark(z0)), MARK(z0))
MARK(h(f(z0))) → c3(ACTIVE(h(active(f(mark(z0))))), H(mark(f(z0))), MARK(f(z0)))
MARK(h(g(z0))) → c3(ACTIVE(h(active(g(z0)))), H(mark(g(z0))), MARK(g(z0)))
MARK(h(h(z0))) → c3(ACTIVE(h(active(h(mark(z0))))), H(mark(h(z0))), MARK(h(z0)))
MARK(h(x0)) → c3

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
ACTIVE(f(x0)) → c(MARK(g(h(f(x0)))), F(x0))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c1(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(g(z0))) → c1(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(h(z0))) → c1(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))
MARK(g(x0)) → c2(G(x0))
MARK(h(z0)) → c3(ACTIVE(h(z0)), H(mark(z0)), MARK(z0))
MARK(h(f(z0))) → c3(ACTIVE(h(active(f(mark(z0))))), H(mark(f(z0))), MARK(f(z0)))
MARK(h(g(z0))) → c3(ACTIVE(h(active(g(z0)))), H(mark(g(z0))), MARK(g(z0)))
MARK(h(h(z0))) → c3(ACTIVE(h(active(h(mark(z0))))), H(mark(h(z0))), MARK(h(z0)))
MARK(h(x0)) → c3

S tuples:

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

K tuples:

MARK(h(z0)) → c3(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c4(F(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

F, G, H, ACTIVE, MARK

Compound Symbols:

c4, c5, c6, c7, c8, c9, c, c1, c2, c3, c3

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

Removed 1 trailing nodes:

MARK(h(x0)) → c3

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
ACTIVE(f(x0)) → c(MARK(g(h(f(x0)))), F(x0))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c1(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(g(z0))) → c1(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(h(z0))) → c1(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))
MARK(g(x0)) → c2(G(x0))
MARK(h(z0)) → c3(ACTIVE(h(z0)), H(mark(z0)), MARK(z0))
MARK(h(f(z0))) → c3(ACTIVE(h(active(f(mark(z0))))), H(mark(f(z0))), MARK(f(z0)))
MARK(h(g(z0))) → c3(ACTIVE(h(active(g(z0)))), H(mark(g(z0))), MARK(g(z0)))
MARK(h(h(z0))) → c3(ACTIVE(h(active(h(mark(z0))))), H(mark(h(z0))), MARK(h(z0)))

S tuples:

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

K tuples:

F(mark(z0)) → c4(F(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

F, G, H, ACTIVE, MARK

Compound Symbols:

c4, c5, c6, c7, c8, c9, c, c1, c2, c3

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

Use narrowing to replace MARK(f(f(z0))) → c1(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0))) by

MARK(f(f(x0))) → c1(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c1(ACTIVE(f(mark(g(h(f(mark(x0))))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c1(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c1(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(g(z0)))) → c1(ACTIVE(f(active(f(active(g(z0)))))), F(mark(f(g(z0)))), MARK(f(g(z0))))
MARK(f(f(h(z0)))) → c1(ACTIVE(f(active(f(active(h(mark(z0))))))), F(mark(f(h(z0)))), MARK(f(h(z0))))
MARK(f(f(x0))) → c1(F(mark(f(x0))))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
ACTIVE(f(x0)) → c(MARK(g(h(f(x0)))), F(x0))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(g(z0))) → c1(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(h(z0))) → c1(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))
MARK(g(x0)) → c2(G(x0))
MARK(h(z0)) → c3(ACTIVE(h(z0)), H(mark(z0)), MARK(z0))
MARK(h(f(z0))) → c3(ACTIVE(h(active(f(mark(z0))))), H(mark(f(z0))), MARK(f(z0)))
MARK(h(g(z0))) → c3(ACTIVE(h(active(g(z0)))), H(mark(g(z0))), MARK(g(z0)))
MARK(h(h(z0))) → c3(ACTIVE(h(active(h(mark(z0))))), H(mark(h(z0))), MARK(h(z0)))
MARK(f(f(x0))) → c1(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c1(ACTIVE(f(mark(g(h(f(mark(x0))))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c1(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c1(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(g(z0)))) → c1(ACTIVE(f(active(f(active(g(z0)))))), F(mark(f(g(z0)))), MARK(f(g(z0))))
MARK(f(f(h(z0)))) → c1(ACTIVE(f(active(f(active(h(mark(z0))))))), F(mark(f(h(z0)))), MARK(f(h(z0))))
MARK(f(f(x0))) → c1(F(mark(f(x0))))

S tuples:

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

K tuples:

F(mark(z0)) → c4(F(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

F, G, H, ACTIVE, MARK

Compound Symbols:

c4, c5, c6, c7, c8, c9, c, c1, c2, c3, c1

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

Use narrowing to replace MARK(f(g(z0))) → c1(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0))) by

MARK(f(g(x0))) → c1(ACTIVE(f(g(x0))), F(mark(g(x0))), MARK(g(x0)))
MARK(f(g(x0))) → c1(F(mark(g(x0))), MARK(g(x0)))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
ACTIVE(f(x0)) → c(MARK(g(h(f(x0)))), F(x0))
MARK(f(z0)) → c1(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(h(z0))) → c1(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))
MARK(g(x0)) → c2(G(x0))
MARK(h(z0)) → c3(ACTIVE(h(z0)), H(mark(z0)), MARK(z0))
MARK(h(f(z0))) → c3(ACTIVE(h(active(f(mark(z0))))), H(mark(f(z0))), MARK(f(z0)))
MARK(h(g(z0))) → c3(ACTIVE(h(active(g(z0)))), H(mark(g(z0))), MARK(g(z0)))
MARK(h(h(z0))) → c3(ACTIVE(h(active(h(mark(z0))))), H(mark(h(z0))), MARK(h(z0)))
MARK(f(f(x0))) → c1(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c1(ACTIVE(f(mark(g(h(f(mark(x0))))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c1(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c1(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(g(z0)))) → c1(ACTIVE(f(active(f(active(g(z0)))))), F(mark(f(g(z0)))), MARK(f(g(z0))))
MARK(f(f(h(z0)))) → c1(ACTIVE(f(active(f(active(h(mark(z0))))))), F(mark(f(h(z0)))), MARK(f(h(z0))))
MARK(f(f(x0))) → c1(F(mark(f(x0))))
MARK(f(g(x0))) → c1(ACTIVE(f(g(x0))), F(mark(g(x0))), MARK(g(x0)))
MARK(f(g(x0))) → c1(F(mark(g(x0))), MARK(g(x0)))

S tuples:

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

K tuples:

F(mark(z0)) → c4(F(z0))
H(mark(z0)) → c8(H(z0))
H(active(z0)) → c9(H(z0))
F(active(z0)) → c5(F(z0))

Defined Rule Symbols:

active, mark, f, g, h

Defined Pair Symbols:

F, G, H, ACTIVE, MARK

Compound Symbols:

c4, c5, c6, c7, c8, c9, c, c1, c2, c3, c1, c1

(33) 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 0.
The certificate found is represented by the following graph.

Start state: 2277
Accept states: [2278, 2279, 2280, 2281, 2282]
Transitions:
2277→2278[active_1|0]
2277→2279[mark_1|0]
2277→2280[f_1|0]
2277→2281[g_1|0]
2277→2282[h_1|0]

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