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(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(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(f(z0))) → mark(c(f(g(f(z0)))))
active(c(z0)) → mark(d(z0))
active(h(z0)) → mark(c(d(z0)))
mark(f(z0)) → active(f(mark(z0)))
mark(c(z0)) → active(c(z0))
mark(g(z0)) → active(g(z0))
mark(d(z0)) → active(d(z0))
mark(h(z0)) → active(h(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
c(mark(z0)) → c(z0)
c(active(z0)) → c(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
d(mark(z0)) → d(z0)
d(active(z0)) → d(z0)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)

Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

S tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

K tuples:none
Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18

(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(g(z0)) → c6(ACTIVE(g(z0)), G(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

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

POL(ACTIVE(x₁)) = 0   
POL(C(x₁)) = 0   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = 0   
POL(MARK(x₁)) = [2]x₁   
POL(active(x₁)) = 0   
POL(c(x₁)) = 0   
POL(c1(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = 0   
POL(f(x₁)) = [2]x₁   
POL(g(x₁)) = [4]   
POL(h(x₁)) = [2]x₁   
POL(mark(x₁)) = [4]x₁   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

S tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18

(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(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

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

POL(ACTIVE(x₁)) = 0   
POL(C(x₁)) = 0   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = 0   
POL(MARK(x₁)) = x₁   
POL(active(x₁)) = [2] + [5]x₁   
POL(c(x₁)) = 0   
POL(c1(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = 0   
POL(f(x₁)) = [2] + x₁   
POL(g(x₁)) = 0   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = 0   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

S tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18

(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.

MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

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

POL(ACTIVE(x₁)) = 0   
POL(C(x₁)) = 0   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = [2]   
POL(MARK(x₁)) = [4]x₁   
POL(active(x₁)) = [1]   
POL(c(x₁)) = 0   
POL(c1(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = 0   
POL(f(x₁)) = [1] + [2]x₁   
POL(g(x₁)) = 0   
POL(h(x₁)) = [2] + [4]x₁   
POL(mark(x₁)) = [1] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

S tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

S tuples:

F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18

(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.

C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

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

POL(ACTIVE(x₁)) = 0   
POL(C(x₁)) = [2]x₁   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = [2]   
POL(MARK(x₁)) = x₁   
POL(active(x₁)) = [1] + [4]x₁   
POL(c(x₁)) = [2]x₁   
POL(c1(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = 0   
POL(f(x₁)) = [4]x₁   
POL(g(x₁)) = 0   
POL(h(x₁)) = [4] + x₁   
POL(mark(x₁)) = [4] + [4]x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

S tuples:

F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18

(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(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

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

POL(ACTIVE(x₁)) = [4]   
POL(C(x₁)) = 0   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = [2] + x₁   
POL(MARK(x₁)) = [4] + [2]x₁   
POL(active(x₁)) = [1] + x₁   
POL(c(x₁)) = 0   
POL(c1(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = 0   
POL(f(x₁)) = [2] + [2]x₁   
POL(g(x₁)) = 0   
POL(h(x₁)) = [4] + [4]x₁   
POL(mark(x₁)) = [1] + [5]x₁   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

S tuples:

F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18

(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.

G(active(z0)) → c14(G(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

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

POL(ACTIVE(x₁)) = [2] + x₁   
POL(C(x₁)) = 0   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = x₁   
POL(H(x₁)) = [1]   
POL(MARK(x₁)) = [3]x₁   
POL(active(x₁)) = [1] + x₁   
POL(c(x₁)) = [2]   
POL(c1(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = [1]   
POL(f(x₁)) = [1] + [4]x₁   
POL(g(x₁)) = [4] + [4]x₁   
POL(h(x₁)) = [4] + [3]x₁   
POL(mark(x₁)) = [2]x₁   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

S tuples:

F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
G(mark(z0)) → c13(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
G(active(z0)) → c14(G(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18

(17) 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)) → c13(G(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

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

POL(ACTIVE(x₁)) = x₁   
POL(C(x₁)) = 0   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = [4]x₁   
POL(H(x₁)) = 0   
POL(MARK(x₁)) = [3]x₁   
POL(active(x₁)) = x₁   
POL(c(x₁)) = [1]   
POL(c1(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = 0   
POL(f(x₁)) = [3] + [4]x₁   
POL(g(x₁)) = [2] + [2]x₁   
POL(h(x₁)) = [3] + [3]x₁   
POL(mark(x₁)) = [1] + [2]x₁   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

S tuples:

F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
G(active(z0)) → c14(G(z0))
G(mark(z0)) → c13(G(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18

(19) 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)) → c9(F(z0))
F(active(z0)) → c10(F(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

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

POL(ACTIVE(x₁)) = x₁   
POL(C(x₁)) = 0   
POL(D(x₁)) = 0   
POL(F(x₁)) = [1] + [2]x₁   
POL(G(x₁)) = 0   
POL(H(x₁)) = [3]   
POL(MARK(x₁)) = [4]x₁   
POL(active(x₁)) = [1] + x₁   
POL(c(x₁)) = 0   
POL(c1(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = 0   
POL(f(x₁)) = [4] + [4]x₁   
POL(g(x₁)) = 0   
POL(h(x₁)) = [2] + [2]x₁   
POL(mark(x₁)) = [1] + [2]x₁   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))

S tuples:

D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
G(active(z0)) → c14(G(z0))
G(mark(z0)) → c13(G(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18

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

Use narrowing to replace ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0)) by

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

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
ACTIVE(f(f(x0))) → c1(MARK(c(f(g(f(x0))))), F(x0))

S tuples:

D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
G(active(z0)) → c14(G(z0))
G(mark(z0)) → c13(G(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1

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

Use narrowing to replace ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0)) by

ACTIVE(h(x0)) → c3(MARK(c(d(x0))), D(x0))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
ACTIVE(f(f(x0))) → c1(MARK(c(f(g(f(x0))))), F(x0))
ACTIVE(h(x0)) → c3(MARK(c(d(x0))), D(x0))

S tuples:

D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
G(active(z0)) → c14(G(z0))
G(mark(z0)) → c13(G(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c2, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1, c3

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

Use narrowing to replace MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0)) by

MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(c(z0))) → c4(ACTIVE(f(active(c(z0)))), F(mark(c(z0))), MARK(c(z0)))
MARK(f(g(z0))) → c4(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(d(z0))) → c4(ACTIVE(f(active(d(z0)))), F(mark(d(z0))), MARK(d(z0)))
MARK(f(h(z0))) → c4(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))
MARK(f(x0)) → c4

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
ACTIVE(f(f(x0))) → c1(MARK(c(f(g(f(x0))))), F(x0))
ACTIVE(h(x0)) → c3(MARK(c(d(x0))), D(x0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(c(z0))) → c4(ACTIVE(f(active(c(z0)))), F(mark(c(z0))), MARK(c(z0)))
MARK(f(g(z0))) → c4(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(d(z0))) → c4(ACTIVE(f(active(d(z0)))), F(mark(d(z0))), MARK(d(z0)))
MARK(f(h(z0))) → c4(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))
MARK(f(x0)) → c4

S tuples:

D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
G(active(z0)) → c14(G(z0))
G(mark(z0)) → c13(G(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c2, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1, c3, c4, c4

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

Removed 1 trailing nodes:

MARK(f(x0)) → c4

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
ACTIVE(f(f(x0))) → c1(MARK(c(f(g(f(x0))))), F(x0))
ACTIVE(h(x0)) → c3(MARK(c(d(x0))), D(x0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(c(z0))) → c4(ACTIVE(f(active(c(z0)))), F(mark(c(z0))), MARK(c(z0)))
MARK(f(g(z0))) → c4(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(d(z0))) → c4(ACTIVE(f(active(d(z0)))), F(mark(d(z0))), MARK(d(z0)))
MARK(f(h(z0))) → c4(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))

S tuples:

D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
G(active(z0)) → c14(G(z0))
G(mark(z0)) → c13(G(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c2, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1, c3, c4

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

Use narrowing to replace MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0)) by

MARK(g(x0)) → c6(G(x0))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
ACTIVE(f(f(x0))) → c1(MARK(c(f(g(f(x0))))), F(x0))
ACTIVE(h(x0)) → c3(MARK(c(d(x0))), D(x0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(c(z0))) → c4(ACTIVE(f(active(c(z0)))), F(mark(c(z0))), MARK(c(z0)))
MARK(f(g(z0))) → c4(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(d(z0))) → c4(ACTIVE(f(active(d(z0)))), F(mark(d(z0))), MARK(d(z0)))
MARK(f(h(z0))) → c4(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))
MARK(g(x0)) → c6(G(x0))

S tuples:

D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
G(active(z0)) → c14(G(z0))
G(mark(z0)) → c13(G(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c2, c5, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1, c3, c4, c6

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

Use narrowing to replace MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0)) by

MARK(d(x0)) → c7(D(x0))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
ACTIVE(f(f(x0))) → c1(MARK(c(f(g(f(x0))))), F(x0))
ACTIVE(h(x0)) → c3(MARK(c(d(x0))), D(x0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(c(z0))) → c4(ACTIVE(f(active(c(z0)))), F(mark(c(z0))), MARK(c(z0)))
MARK(f(g(z0))) → c4(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(d(z0))) → c4(ACTIVE(f(active(d(z0)))), F(mark(d(z0))), MARK(d(z0)))
MARK(f(h(z0))) → c4(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))
MARK(g(x0)) → c6(G(x0))
MARK(d(x0)) → c7(D(x0))

S tuples:

D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
G(active(z0)) → c14(G(z0))
G(mark(z0)) → c13(G(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c2, c5, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1, c3, c4, c6, c7

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

Use narrowing to replace MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0)) by

MARK(h(z0)) → c8(ACTIVE(h(z0)), H(mark(z0)), MARK(z0))
MARK(h(f(z0))) → c8(ACTIVE(h(active(f(mark(z0))))), H(mark(f(z0))), MARK(f(z0)))
MARK(h(c(z0))) → c8(ACTIVE(h(active(c(z0)))), H(mark(c(z0))), MARK(c(z0)))
MARK(h(g(z0))) → c8(ACTIVE(h(active(g(z0)))), H(mark(g(z0))), MARK(g(z0)))
MARK(h(d(z0))) → c8(ACTIVE(h(active(d(z0)))), H(mark(d(z0))), MARK(d(z0)))
MARK(h(h(z0))) → c8(ACTIVE(h(active(h(mark(z0))))), H(mark(h(z0))), MARK(h(z0)))
MARK(h(x0)) → c8

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
ACTIVE(f(f(x0))) → c1(MARK(c(f(g(f(x0))))), F(x0))
ACTIVE(h(x0)) → c3(MARK(c(d(x0))), D(x0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(c(z0))) → c4(ACTIVE(f(active(c(z0)))), F(mark(c(z0))), MARK(c(z0)))
MARK(f(g(z0))) → c4(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(d(z0))) → c4(ACTIVE(f(active(d(z0)))), F(mark(d(z0))), MARK(d(z0)))
MARK(f(h(z0))) → c4(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))
MARK(g(x0)) → c6(G(x0))
MARK(d(x0)) → c7(D(x0))
MARK(h(z0)) → c8(ACTIVE(h(z0)), H(mark(z0)), MARK(z0))
MARK(h(f(z0))) → c8(ACTIVE(h(active(f(mark(z0))))), H(mark(f(z0))), MARK(f(z0)))
MARK(h(c(z0))) → c8(ACTIVE(h(active(c(z0)))), H(mark(c(z0))), MARK(c(z0)))
MARK(h(g(z0))) → c8(ACTIVE(h(active(g(z0)))), H(mark(g(z0))), MARK(g(z0)))
MARK(h(d(z0))) → c8(ACTIVE(h(active(d(z0)))), H(mark(d(z0))), MARK(d(z0)))
MARK(h(h(z0))) → c8(ACTIVE(h(active(h(mark(z0))))), H(mark(h(z0))), MARK(h(z0)))
MARK(h(x0)) → c8

S tuples:

D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))

K tuples:

MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
G(active(z0)) → c14(G(z0))
G(mark(z0)) → c13(G(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c2, c5, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1, c3, c4, c6, c7, c8, c8

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

Removed 1 trailing nodes:

MARK(h(x0)) → c8

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
ACTIVE(f(f(x0))) → c1(MARK(c(f(g(f(x0))))), F(x0))
ACTIVE(h(x0)) → c3(MARK(c(d(x0))), D(x0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(c(z0))) → c4(ACTIVE(f(active(c(z0)))), F(mark(c(z0))), MARK(c(z0)))
MARK(f(g(z0))) → c4(ACTIVE(f(active(g(z0)))), F(mark(g(z0))), MARK(g(z0)))
MARK(f(d(z0))) → c4(ACTIVE(f(active(d(z0)))), F(mark(d(z0))), MARK(d(z0)))
MARK(f(h(z0))) → c4(ACTIVE(f(active(h(mark(z0))))), F(mark(h(z0))), MARK(h(z0)))
MARK(g(x0)) → c6(G(x0))
MARK(d(x0)) → c7(D(x0))
MARK(h(z0)) → c8(ACTIVE(h(z0)), H(mark(z0)), MARK(z0))
MARK(h(f(z0))) → c8(ACTIVE(h(active(f(mark(z0))))), H(mark(f(z0))), MARK(f(z0)))
MARK(h(c(z0))) → c8(ACTIVE(h(active(c(z0)))), H(mark(c(z0))), MARK(c(z0)))
MARK(h(g(z0))) → c8(ACTIVE(h(active(g(z0)))), H(mark(g(z0))), MARK(g(z0)))
MARK(h(d(z0))) → c8(ACTIVE(h(active(d(z0)))), H(mark(d(z0))), MARK(d(z0)))
MARK(h(h(z0))) → c8(ACTIVE(h(active(h(mark(z0))))), H(mark(h(z0))), MARK(h(z0)))

S tuples:

D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))

K tuples:

MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
G(active(z0)) → c14(G(z0))
G(mark(z0)) → c13(G(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))

Defined Rule Symbols:

active, mark, f, c, g, d, h

Defined Pair Symbols:

ACTIVE, MARK, F, C, G, D, H

Compound Symbols:

c2, c5, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1, c3, c4, c6, c7, c8

(37) 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: 3931
Accept states: [3932, 3933, 3934, 3935, 3936, 3937, 3938]
Transitions:
3931→3932[active_1|0]
3931→3933[mark_1|0]
3931→3934[f_1|0]
3931→3935[c_1|0]
3931→3936[g_1|0]
3931→3937[d_1|0]
3931→3938[h_1|0]

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