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)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(z0))) → mark(c(f(g(f(z0)))))
active(c(z0)) → mark(d(z0))
active(h(z0)) → mark(c(d(z0)))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(c(z0)) → c(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(d(z0)) → d(proper(z0))
proper(h(z0)) → h(proper(z0))
c(ok(z0)) → ok(c(z0))
g(ok(z0)) → ok(g(z0))
d(ok(z0)) → ok(d(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, f, h, proper, c, g, d, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, C, G, D, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

(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.

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(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(PROPER(x₁)) = 0   
POL(TOP(x₁)) = [4]x₁   
POL(active(x₁)) = [4]   
POL(c(x₁)) = [4]x₁   
POL(c1(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c13(x₁, x₂)) = x₁ + x₂   
POL(c14(x₁, x₂)) = x₁ + x₂   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c19(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = [4]x₁   
POL(f(x₁)) = x₁   
POL(g(x₁)) = x₁   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = [4]   
POL(ok(x₁)) = [4]   
POL(proper(x₁)) = 0   

(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)))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(c(z0)) → c(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(d(z0)) → d(proper(z0))
proper(h(z0)) → h(proper(z0))
c(ok(z0)) → ok(c(z0))
g(ok(z0)) → ok(g(z0))
d(ok(z0)) → ok(d(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, c, g, d, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, C, G, D, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

(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.

C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

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

POL(ACTIVE(x₁)) = [2]x₁   
POL(C(x₁)) = x₁   
POL(D(x₁)) = [4]x₁   
POL(F(x₁)) = 0   
POL(G(x₁)) = [2]x₁   
POL(H(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = [2]x₁   
POL(active(x₁)) = 0   
POL(c(x₁)) = [4]x₁   
POL(c1(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c13(x₁, x₂)) = x₁ + x₂   
POL(c14(x₁, x₂)) = x₁ + x₂   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c19(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = [4]x₁   
POL(f(x₁)) = [2]x₁   
POL(g(x₁)) = x₁   
POL(h(x₁)) = [4]x₁   
POL(mark(x₁)) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(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)))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(c(z0)) → c(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(d(z0)) → d(proper(z0))
proper(h(z0)) → h(proper(z0))
c(ok(z0)) → ok(c(z0))
g(ok(z0)) → ok(g(z0))
d(ok(z0)) → ok(d(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

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

K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, f, h, proper, c, g, d, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, C, G, D, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

(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.

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

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

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

POL(ACTIVE(x₁)) = [1]   
POL(C(x₁)) = 0   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = x₁   
POL(active(x₁)) = 0   
POL(c(x₁)) = [2]x₁   
POL(c1(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c13(x₁, x₂)) = x₁ + x₂   
POL(c14(x₁, x₂)) = x₁ + x₂   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c19(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = [4]x₁   
POL(f(x₁)) = [2]x₁   
POL(g(x₁)) = [4]x₁   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = 0   
POL(ok(x₁)) = [1]   
POL(proper(x₁)) = 0   

(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)))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(c(z0)) → c(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(d(z0)) → d(proper(z0))
proper(h(z0)) → h(proper(z0))
c(ok(z0)) → ok(c(z0))
g(ok(z0)) → ok(g(z0))
d(ok(z0)) → ok(d(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))

K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))

Defined Rule Symbols:

active, f, h, proper, c, g, d, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, C, G, D, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

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

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

F(ok(z0)) → c7(F(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(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)) = [2]x₁   
POL(G(x₁)) = 0   
POL(H(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = x₁²   
POL(active(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c13(x₁, x₂)) = x₁ + x₂   
POL(c14(x₁, x₂)) = x₁ + x₂   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c19(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = x₁   
POL(f(x₁)) = [3]x₁   
POL(g(x₁)) = x₁   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = 0   

(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)))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(c(z0)) → c(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(d(z0)) → d(proper(z0))
proper(h(z0)) → h(proper(z0))
c(ok(z0)) → ok(c(z0))
g(ok(z0)) → ok(g(z0))
d(ok(z0)) → ok(d(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))

K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
F(ok(z0)) → c7(F(z0))

Defined Rule Symbols:

active, f, h, proper, c, g, d, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, C, G, D, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

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

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

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

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

POL(ACTIVE(x₁)) = [3] + [2]x₁   
POL(C(x₁)) = 0   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = [2]x₁²   
POL(active(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c13(x₁, x₂)) = x₁ + x₂   
POL(c14(x₁, x₂)) = x₁ + x₂   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c19(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = x₁   
POL(f(x₁)) = [1] + [2]x₁   
POL(g(x₁)) = x₁   
POL(h(x₁)) = [2]x₁   
POL(mark(x₁)) = x₁   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = 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)))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(c(z0)) → c(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(d(z0)) → d(proper(z0))
proper(h(z0)) → h(proper(z0))
c(ok(z0)) → ok(c(z0))
g(ok(z0)) → ok(g(z0))
d(ok(z0)) → ok(d(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))

K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, f, h, proper, c, g, d, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, C, G, D, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

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

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

PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(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(PROPER(x₁)) = [1] + x₁   
POL(TOP(x₁)) = x₁ + x₁²   
POL(active(x₁)) = [2] + x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c13(x₁, x₂)) = x₁ + x₂   
POL(c14(x₁, x₂)) = x₁ + x₂   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c19(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = x₁   
POL(f(x₁)) = x₁   
POL(g(x₁)) = [1] + x₁   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = 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)))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(c(z0)) → c(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(d(z0)) → d(proper(z0))
proper(h(z0)) → h(proper(z0))
c(ok(z0)) → ok(c(z0))
g(ok(z0)) → ok(g(z0))
d(ok(z0)) → ok(d(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))

K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, c, g, d, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, C, G, D, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

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

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

PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(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(PROPER(x₁)) = [1] + x₁   
POL(TOP(x₁)) = [2]x₁²   
POL(active(x₁)) = [3] + x₁   
POL(c(x₁)) = [2] + x₁   
POL(c1(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c13(x₁, x₂)) = x₁ + x₂   
POL(c14(x₁, x₂)) = x₁ + x₂   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c19(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = x₁   
POL(f(x₁)) = x₁   
POL(g(x₁)) = x₁   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = [3] + x₁   
POL(proper(x₁)) = 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)))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(c(z0)) → c(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(d(z0)) → d(proper(z0))
proper(h(z0)) → h(proper(z0))
c(ok(z0)) → ok(c(z0))
g(ok(z0)) → ok(g(z0))
d(ok(z0)) → ok(d(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))

K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, c, g, d, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, C, G, D, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

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

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

H(ok(z0)) → c9(H(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

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

POL(ACTIVE(x₁)) = x₁   
POL(C(x₁)) = x₁   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = [2]x₁²   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = x₁²   
POL(active(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c13(x₁, x₂)) = x₁ + x₂   
POL(c14(x₁, x₂)) = x₁ + x₂   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c19(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = x₁   
POL(f(x₁)) = x₁   
POL(g(x₁)) = x₁   
POL(h(x₁)) = x₁ + [2]x₁²   
POL(mark(x₁)) = x₁   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   

(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)))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(c(z0)) → c(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(d(z0)) → d(proper(z0))
proper(h(z0)) → h(proper(z0))
c(ok(z0)) → ok(c(z0))
g(ok(z0)) → ok(g(z0))
d(ok(z0)) → ok(d(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))

K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
H(ok(z0)) → c9(H(z0))

Defined Rule Symbols:

active, f, h, proper, c, g, d, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, C, G, D, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

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

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

ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

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

POL(ACTIVE(x₁)) = [3]x₁   
POL(C(x₁)) = 0   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = [1]   
POL(PROPER(x₁)) = [1] + [2]x₁   
POL(TOP(x₁)) = x₁ + [2]x₁²   
POL(active(x₁)) = [1] + x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c13(x₁, x₂)) = x₁ + x₂   
POL(c14(x₁, x₂)) = x₁ + x₂   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c19(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = x₁   
POL(f(x₁)) = x₁   
POL(g(x₁)) = x₁   
POL(h(x₁)) = [1] + x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = 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)))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(c(z0)) → c(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(d(z0)) → d(proper(z0))
proper(h(z0)) → h(proper(z0))
c(ok(z0)) → ok(c(z0))
g(ok(z0)) → ok(g(z0))
d(ok(z0)) → ok(d(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))

K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
H(ok(z0)) → c9(H(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, c, g, d, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, C, G, D, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

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

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

PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(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(PROPER(x₁)) = [2]x₁   
POL(TOP(x₁)) = [2]x₁²   
POL(active(x₁)) = [2] + x₁   
POL(c(x₁)) = [1] + x₁   
POL(c1(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c13(x₁, x₂)) = x₁ + x₂   
POL(c14(x₁, x₂)) = x₁ + x₂   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c19(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(d(x₁)) = x₁   
POL(f(x₁)) = [1] + x₁   
POL(g(x₁)) = x₁   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = x₁   

(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)))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(c(z0)) → c(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(d(z0)) → d(proper(z0))
proper(h(z0)) → h(proper(z0))
c(ok(z0)) → ok(c(z0))
g(ok(z0)) → ok(g(z0))
d(ok(z0)) → ok(d(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))

K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
H(ok(z0)) → c9(H(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, c, g, d, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, C, G, D, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

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

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

ACTIVE(f(f(x0))) → c1(F(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)))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(c(z0)) → c(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(d(z0)) → d(proper(z0))
proper(h(z0)) → h(proper(z0))
c(ok(z0)) → ok(c(z0))
g(ok(z0)) → ok(g(z0))
d(ok(z0)) → ok(d(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
F(ok(z0)) → c7(F(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(x0))) → c1(F(x0))

S tuples:

F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))

K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
D(ok(z0)) → c17(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(h(z0)) → c3(C(d(z0)), D(z0))
F(ok(z0)) → c7(F(z0))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c12(G(proper(z0)), PROPER(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
H(ok(z0)) → c9(H(z0))
ACTIVE(h(z0)) → c5(H(active(z0)), ACTIVE(z0))
PROPER(h(z0)) → c14(H(proper(z0)), PROPER(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, c, g, d, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, C, G, D, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c1

(25) CpxTrsMatchBoundsProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1.
The certificate found is represented by the following graph.

Start state: 3252
Accept states: [3253, 3254, 3255, 3256, 3257, 3258, 3259, 3260]
Transitions:
3252→3253[active_1|0]
3252→3254[f_1|0]
3252→3255[h_1|0]
3252→3256[proper_1|0]
3252→3257[c_1|0]
3252→3258[g_1|0]
3252→3259[d_1|0]
3252→3260[top_1|0]
3252→3252[mark_1|0, ok_1|0]
3252→3261[f_1|1]
3252→3262[h_1|1]
3252→3263[proper_1|1]
3252→3264[f_1|1]
3252→3265[h_1|1]
3252→3266[c_1|1]
3252→3267[g_1|1]
3252→3268[d_1|1]
3252→3269[active_1|1]
3261→3254[mark_1|1]
3261→3261[mark_1|1]
3261→3264[mark_1|1]
3262→3255[mark_1|1]
3262→3262[mark_1|1]
3262→3265[mark_1|1]
3263→3260[top_1|1]
3264→3254[ok_1|1]
3264→3261[ok_1|1]
3264→3264[ok_1|1]
3265→3255[ok_1|1]
3265→3262[ok_1|1]
3265→3265[ok_1|1]
3266→3257[ok_1|1]
3266→3266[ok_1|1]
3267→3258[ok_1|1]
3267→3267[ok_1|1]
3268→3259[ok_1|1]
3268→3268[ok_1|1]
3269→3260[top_1|1]

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