The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
    ↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 564 ms)
    ↳4 CdtProblem
    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 557 ms)
    ↳6 CdtProblem
    ↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 312 ms)
    ↳8 CdtProblem
    ↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 945 ms)
    ↳10 CdtProblem
    ↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 1128 ms)
    ↳12 CdtProblem
    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 259 ms)
    ↳14 CdtProblem
    ↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 314 ms)
    ↳16 CdtProblem
    ↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 844 ms)
    ↳18 CdtProblem
    ↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 2361 ms)
    ↳20 CdtProblem
    ↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳22 CdtProblem
    ↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳24 CdtProblem
    ↳25 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳26 CdtProblem
    ↳27 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳28 CdtProblem
    ↳29 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳30 CdtProblem
    ↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳32 CdtProblem
    ↳33 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳34 CdtProblem
    ↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳36 CdtProblem
    ↳37 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳38 CdtProblem
    ↳39 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳40 CdtProblem
    ↳41 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳42 CdtProblem
↳43 CpxTrsMatchBoundsProof (⇔, 0 ms)
↳44 BOUNDS(O(1), O(n^1))

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
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(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(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(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

(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.

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = [2]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, x2, x3)) = x1 + x2 + x3   
POL(c1(x1, x2)) = x1 + x2   
POL(c10(x1)) = x1   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2)) = x1 + x2   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1)) = x1   
POL(g(x1)) = [4]x1   
POL(h(x1)) = x1   
POL(mark(x1)) = 0   
POL(ok(x1)) = [2] + x1   
POL(proper(x1)) = 0   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

(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.

TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0   
POL(F(x1)) = 0   
POL(G(x1)) = 0   
POL(H(x1)) = 0   
POL(PROPER(x1)) = [3]   
POL(TOP(x1)) = [4]x1   
POL(active(x1)) = [2]   
POL(c(x1, x2, x3)) = x1 + x2 + x3   
POL(c1(x1, x2)) = x1 + x2   
POL(c10(x1)) = x1   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2)) = x1 + x2   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1)) = x1   
POL(g(x1)) = [2]x1   
POL(h(x1)) = x1   
POL(mark(x1)) = [1]   
POL(ok(x1)) = [2] + x1   
POL(proper(x1)) = 0   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

(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(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = [1]   
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, x2, x3)) = x1 + x2 + x3   
POL(c1(x1, x2)) = x1 + x2   
POL(c10(x1)) = x1   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2)) = x1 + x2   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1)) = x1   
POL(g(x1)) = [4]x1   
POL(h(x1)) = [4]x1   
POL(mark(x1)) = 0   
POL(ok(x1)) = [1]   
POL(proper(x1)) = 0   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

(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.

PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0   
POL(F(x1)) = 0   
POL(G(x1)) = 0   
POL(H(x1)) = 0   
POL(PROPER(x1)) = [2] + [2]x1   
POL(TOP(x1)) = [2]x12   
POL(active(x1)) = [3] + x1   
POL(c(x1, x2, x3)) = x1 + x2 + x3   
POL(c1(x1, x2)) = x1 + x2   
POL(c10(x1)) = x1   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2)) = x1 + x2   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1)) = [1] + x1   
POL(g(x1)) = x1   
POL(h(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = [3] + x1   
POL(proper(x1)) = x1   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

(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.

PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0   
POL(F(x1)) = 0   
POL(G(x1)) = 0   
POL(H(x1)) = 0   
POL(PROPER(x1)) = x1   
POL(TOP(x1)) = x12   
POL(active(x1)) = [3] + x1   
POL(c(x1, x2, x3)) = x1 + x2 + x3   
POL(c1(x1, x2)) = x1 + x2   
POL(c10(x1)) = x1   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2)) = x1 + x2   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1)) = x1   
POL(g(x1)) = x1   
POL(h(x1)) = [2] + x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = [3] + x1   
POL(proper(x1)) = x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

(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.

F(ok(z0)) → c4(F(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = x1   
POL(F(x1)) = [2]x12   
POL(G(x1)) = 0   
POL(H(x1)) = 0   
POL(PROPER(x1)) = x1 + x12   
POL(TOP(x1)) = x1 + x12   
POL(active(x1)) = x1   
POL(c(x1, x2, x3)) = x1 + x2 + x3   
POL(c1(x1, x2)) = x1 + x2   
POL(c10(x1)) = x1   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2)) = x1 + x2   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1)) = x1 + [3]x12   
POL(g(x1)) = x1   
POL(h(x1)) = x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = [2] + x1   
POL(proper(x1)) = 0   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

(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.

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = [2] + [2]x1   
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, x2, x3)) = x1 + x2 + x3   
POL(c1(x1, x2)) = x1 + x2   
POL(c10(x1)) = x1   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2)) = x1 + x2   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1)) = [1] + [2]x1   
POL(g(x1)) = x1   
POL(h(x1)) = x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = [2] + x1   
POL(proper(x1)) = x1   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

(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.

ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = [1] + [2]x1   
POL(F(x1)) = 0   
POL(G(x1)) = 0   
POL(H(x1)) = [1]   
POL(PROPER(x1)) = [1] + x1   
POL(TOP(x1)) = [2]x12   
POL(active(x1)) = [2] + x1   
POL(c(x1, x2, x3)) = x1 + x2 + x3   
POL(c1(x1, x2)) = x1 + x2   
POL(c10(x1)) = x1   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2)) = x1 + x2   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1)) = x1   
POL(g(x1)) = x1   
POL(h(x1)) = [1] + x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = [3] + x1   
POL(proper(x1)) = x1   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

(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.

PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0   
POL(F(x1)) = 0   
POL(G(x1)) = 0   
POL(H(x1)) = 0   
POL(PROPER(x1)) = [2]x1   
POL(TOP(x1)) = [2]x12   
POL(active(x1)) = [2] + x1   
POL(c(x1, x2, x3)) = x1 + x2 + x3   
POL(c1(x1, x2)) = x1 + x2   
POL(c10(x1)) = x1   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2)) = x1 + x2   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
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   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

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

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

ACTIVE(f(x0)) → c(F(x0))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0)) → c(F(x0))
S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

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

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

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

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0)) → c(F(x0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(f(x0)) → c1
S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c, c1, c1

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

Removed 1 trailing nodes:

ACTIVE(f(x0)) → c1

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0)) → c(F(x0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c, c1

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

Use narrowing to replace ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) by

ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(x0)) → c2

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0)) → c(F(x0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(x0)) → c2
S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c, c1, c2, c2

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

Removed 1 trailing nodes:

ACTIVE(h(x0)) → c2

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0)) → c(F(x0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c, c1, c2

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

Use narrowing to replace PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) by

PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(f(x0)) → c7

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0)) → c(F(x0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(f(x0)) → c7
S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c8, c9, c10, c11, c12, c, c1, c2, c7, c7

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

Removed 1 trailing nodes:

PROPER(f(x0)) → c7

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0)) → c(F(x0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c8, c9, c10, c11, c12, c, c1, c2, c7

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

Use narrowing to replace PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) by

PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(x0)) → c8

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0)) → c(F(x0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(x0)) → c8
S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c9, c10, c11, c12, c, c1, c2, c7, c8, c8

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

Removed 1 trailing nodes:

PROPER(g(x0)) → c8

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0)) → c(F(x0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))
S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c9, c10, c11, c12, c, c1, c2, c7, c8

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

Use narrowing to replace PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) by

PROPER(h(f(z0))) → c9(H(f(proper(z0))), PROPER(f(z0)))
PROPER(h(g(z0))) → c9(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c9(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(x0)) → c9

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0)) → c(F(x0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))
PROPER(h(f(z0))) → c9(H(f(proper(z0))), PROPER(f(z0)))
PROPER(h(g(z0))) → c9(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c9(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(x0)) → c9
S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, G, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c5, c6, c10, c11, c12, c, c1, c2, c7, c8, c9, c9

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

Removed 1 trailing nodes:

PROPER(h(x0)) → c9

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(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(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0)) → c(F(x0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))
PROPER(h(f(z0))) → c9(H(f(proper(z0))), PROPER(f(z0)))
PROPER(h(g(z0))) → c9(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c9(H(h(proper(z0))), PROPER(h(z0)))
S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, G, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c5, c6, c10, c11, c12, c, c1, c2, c7, c8, c9

(43) 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: 3240
Accept states: [3241, 3242, 3243, 3244, 3245, 3246]
Transitions:
3240→3241[active_1|0]
3240→3242[f_1|0]
3240→3243[h_1|0]
3240→3244[proper_1|0]
3240→3245[g_1|0]
3240→3246[top_1|0]
3240→3240[mark_1|0, ok_1|0]
3240→3247[f_1|1]
3240→3248[h_1|1]
3240→3249[proper_1|1]
3240→3250[f_1|1]
3240→3251[h_1|1]
3240→3252[g_1|1]
3240→3253[active_1|1]
3247→3242[mark_1|1]
3247→3247[mark_1|1]
3247→3250[mark_1|1]
3248→3243[mark_1|1]
3248→3248[mark_1|1]
3248→3251[mark_1|1]
3249→3246[top_1|1]
3250→3242[ok_1|1]
3250→3247[ok_1|1]
3250→3250[ok_1|1]
3251→3243[ok_1|1]
3251→3248[ok_1|1]
3251→3251[ok_1|1]
3252→3245[ok_1|1]
3252→3252[ok_1|1]
3253→3246[top_1|1]

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