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) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))

S tuples:

ACTIVE(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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:

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

(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))

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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))

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

POL(ACTIVE(x₁)) = 0   
POL(C(x₁)) = [4]x₁   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = x₁   
POL(H(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = 0   
POL(active(x₁)) = [3] + [2]x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
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)) = x₁   
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)) = [2]x₁   
POL(g(x₁)) = [3]x₁   
POL(h(x₁)) = [4]x₁   
POL(mark(x₁)) = 0   
POL(ok(x₁)) = [2] + 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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))

S tuples:

ACTIVE(c(z0)) → c2(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))
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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))

K tuples:

C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(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, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c1, c3

(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(c(z0)) → c2(D(z0))
ACTIVE(f(f(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))

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

POL(ACTIVE(x₁)) = [2]   
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]x₁   
POL(active(x₁)) = [4]   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
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)) = x₁   
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)) = [4]   
POL(ok(x₁)) = [5]   
POL(proper(x₁)) = [4]   

(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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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))
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:

C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(f(f(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(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:

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

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

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

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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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)) = [2]x₁   
POL(active(x₁)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
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)) = x₁   
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)) = [2]x₁   
POL(f(x₁)) = [4]x₁   
POL(g(x₁)) = x₁   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = 0   
POL(ok(x₁)) = [1]   
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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

K tuples:

C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(f(f(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(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:

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

(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.

D(ok(z0)) → c17(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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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)) = x₁   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = [4]x₁   
POL(active(x₁)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
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)) = x₁   
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)) = [3]x₁   
POL(f(x₁)) = [4]x₁   
POL(g(x₁)) = [2]x₁   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   

(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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

K tuples:

C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(f(f(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
D(ok(z0)) → c17(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:

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

(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.

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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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)) = x₁   
POL(active(x₁)) = x₁   
POL(c(x₁)) = [1] + [3]x₁   
POL(c1(x₁)) = x₁   
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)) = x₁   
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)) = 0   
POL(f(x₁)) = [1] + [4]x₁   
POL(g(x₁)) = 0   
POL(h(x₁)) = [4] + x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = 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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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:

C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(f(f(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
D(ok(z0)) → c17(D(z0))
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:

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

(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.

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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))

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

POL(ACTIVE(x₁)) = [1] + [2]x₁   
POL(C(x₁)) = 0   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(H(x₁)) = x₁ + [3]x₁²   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = [2]x₁²   
POL(active(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
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)) = x₁   
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)) = [2]x₁   
POL(g(x₁)) = x₁   
POL(h(x₁)) = [2]x₁ + [2]x₁²   
POL(mark(x₁)) = x₁   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   

(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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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))
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:

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

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

(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.

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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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)) = [1] + x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
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)) = x₁   
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)) = [1] + x₁   
POL(proper(x₁)) = 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)))
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(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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))
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:

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

(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.

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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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)) = [2]x₁²   
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)) = x₁   
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)) = x₁   
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)) = [2]x₁ + [2]x₁²   
POL(g(x₁)) = x₁   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   

(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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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))
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:

C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(f(f(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(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:

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

(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.

ACTIVE(h(z0)) → c5(H(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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))

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

POL(ACTIVE(x₁)) = [1] + [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)) = x₁²   
POL(active(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
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)) = x₁   
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)) = [2]x₁   
POL(g(x₁)) = x₁   
POL(h(x₁)) = [1] + [2]x₁   
POL(mark(x₁)) = x₁   
POL(ok(x₁)) = [1] + 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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))

S tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(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:

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

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

(23) 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(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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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)) = x₁   
POL(TOP(x₁)) = [2]x₁²   
POL(active(x₁)) = [2] + x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
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)) = x₁   
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)) = [2] + x₁   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = x₁   

(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(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))

S tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(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))

K tuples:

C(ok(z0)) → c15(C(z0))
G(ok(z0)) → c16(G(z0))
ACTIVE(c(z0)) → c2(D(z0))
ACTIVE(f(f(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
D(ok(z0)) → c17(D(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
H(ok(z0)) → c9(H(z0))
PROPER(f(z0)) → c10(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c7(F(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:

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

(25) 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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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)) = 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)) = x₁   
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)) = x₁   
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)) = x₁   
POL(mark(x₁)) = x₁   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = x₁   

(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)))
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(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))

S tuples:

F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(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))

K tuples:

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

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

(27) 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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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]x₁ + x₁²   
POL(active(x₁)) = [3] + x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
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)) = x₁   
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)) = [2] + x₁   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = [3] + x₁   
POL(proper(x₁)) = x₁   

(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)))
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(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))

S tuples:

F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))
PROPER(d(z0)) → c13(D(proper(z0)), PROPER(z0))

K tuples:

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

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

(29) 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(d(z0)) → c13(D(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(c(z0)) → c2(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(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] + [2]x₁   
POL(TOP(x₁)) = [2]x₁ + [2]x₁²   
POL(active(x₁)) = [2] + x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
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)) = x₁   
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)) = [1] + x₁   
POL(f(x₁)) = x₁   
POL(g(x₁)) = x₁   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = x₁   

(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)))
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(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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))

S tuples:

F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))

K tuples:

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

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

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

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

(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)))
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)) → 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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))
ACTIVE(f(f(f(z0)))) → c4(F(mark(c(f(g(f(z0)))))), ACTIVE(f(f(z0))))
ACTIVE(f(c(z0))) → c4(F(mark(d(z0))), ACTIVE(c(z0)))
ACTIVE(f(h(z0))) → c4(F(mark(c(d(z0)))), ACTIVE(h(z0)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c4(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(f(x0)) → c4

S tuples:

F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))

K tuples:

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

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

Removed 1 trailing nodes:

ACTIVE(f(x0)) → c4

(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)))
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)) → 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(z0))) → c1(F(z0))
ACTIVE(h(z0)) → c3(D(z0))
ACTIVE(f(f(f(z0)))) → c4(F(mark(c(f(g(f(z0)))))), ACTIVE(f(f(z0))))
ACTIVE(f(c(z0))) → c4(F(mark(d(z0))), ACTIVE(c(z0)))
ACTIVE(f(h(z0))) → c4(F(mark(c(d(z0)))), ACTIVE(h(z0)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c4(F(h(active(z0))), ACTIVE(h(z0)))

S tuples:

F(mark(z0)) → c6(F(z0))
H(mark(z0)) → c8(H(z0))
PROPER(c(z0)) → c11(C(proper(z0)), PROPER(z0))

K tuples:

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

(35) 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: 3926
Accept states: [3927, 3928, 3929, 3930, 3931, 3932, 3933, 3934]
Transitions:
3926→3927[active_1|0]
3926→3928[f_1|0]
3926→3929[h_1|0]
3926→3930[proper_1|0]
3926→3931[c_1|0]
3926→3932[g_1|0]
3926→3933[d_1|0]
3926→3934[top_1|0]
3926→3926[mark_1|0, ok_1|0]
3926→3935[f_1|1]
3926→3936[h_1|1]
3926→3937[proper_1|1]
3926→3938[f_1|1]
3926→3939[h_1|1]
3926→3940[c_1|1]
3926→3941[g_1|1]
3926→3942[d_1|1]
3926→3943[active_1|1]
3935→3928[mark_1|1]
3935→3935[mark_1|1]
3935→3938[mark_1|1]
3936→3929[mark_1|1]
3936→3936[mark_1|1]
3936→3939[mark_1|1]
3937→3934[top_1|1]
3938→3928[ok_1|1]
3938→3935[ok_1|1]
3938→3938[ok_1|1]
3939→3929[ok_1|1]
3939→3936[ok_1|1]
3939→3939[ok_1|1]
3940→3931[ok_1|1]
3940→3940[ok_1|1]
3941→3932[ok_1|1]
3941→3941[ok_1|1]
3942→3933[ok_1|1]
3942→3942[ok_1|1]
3943→3934[top_1|1]

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