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(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(f(a))) → c(F(g(f(a))), G(f(a)), F(a))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(a) → c5
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(f(a))) → c(F(g(f(a))), G(f(a)), F(a))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(a) → c5
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

ACTIVE, F, PROPER, G, TOP

Compound Symbols:

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

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

Removed 2 trailing nodes:

ACTIVE(f(f(a))) → c(F(g(f(a))), G(f(a)), F(a))
PROPER(a) → c5

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(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))
F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

ACTIVE, F, PROPER, G, TOP

Compound Symbols:

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

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

ACTIVE, F, PROPER, G, TOP

Compound Symbols:

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

(7) CdtRuleRemovalProof (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)) → c8(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

g(ok(z0)) → ok(g(z0))
active(f(f(a))) → mark(f(g(f(a))))
proper(g(z0)) → g(proper(z0))
active(f(z0)) → f(active(z0))
f(ok(z0)) → ok(f(z0))
f(mark(z0)) → mark(f(z0))
proper(a) → ok(a)
proper(f(z0)) → f(proper(z0))

And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(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(PROPER(x₁)) = [1]   
POL(TOP(x₁)) = [2]x₁   
POL(a) = [2]   
POL(active(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(f(x₁)) = x₁   
POL(g(x₁)) = [1]   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))

K tuples:

TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

ACTIVE, F, PROPER, G, TOP

Compound Symbols:

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

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

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

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))

K tuples:

TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

F, PROPER, G, TOP, ACTIVE

Compound Symbols:

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

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

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

ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))

We considered the (Usable) Rules:

g(ok(z0)) → ok(g(z0))
active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
f(mark(z0)) → mark(f(z0))
proper(a) → ok(a)
proper(f(z0)) → f(proper(z0))

And the Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))

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

POL(ACTIVE(x₁)) = x₁   
POL(F(x₁)) = 0   
POL(G(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = [4]x₁   
POL(a) = [2]   
POL(active(x₁)) = 0   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(f(x₁)) = [4]x₁   
POL(g(x₁)) = 0   
POL(mark(x₁)) = x₁   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))

K tuples:

TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))

Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

F, PROPER, G, TOP, ACTIVE

Compound Symbols:

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

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

Use narrowing to replace PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) by

PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(a)) → c4(F(ok(a)), PROPER(a))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))

K tuples:

TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))

Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

F, PROPER, G, TOP, ACTIVE

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))

K tuples:

TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))

Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

F, PROPER, G, TOP, ACTIVE

Compound Symbols:

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

(17) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

PROPER(f(a)) → c4(F(ok(a)))

We considered the (Usable) Rules:

g(ok(z0)) → ok(g(z0))
active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
f(mark(z0)) → mark(f(z0))
proper(a) → ok(a)
proper(f(z0)) → f(proper(z0))

And the Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))

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(PROPER(x₁)) = [1]   
POL(TOP(x₁)) = x₁   
POL(a) = [2]   
POL(active(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(f(x₁)) = [4]x₁   
POL(g(x₁)) = [1]   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = x₁   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0))
G(ok(z0)) → c7(G(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))

K tuples:

TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
PROPER(f(a)) → c4(F(ok(a)))

Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

F, PROPER, G, TOP, ACTIVE

Compound Symbols:

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

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

Use narrowing to replace PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) by

PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(a)) → c6(G(ok(a)), PROPER(a))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))

K tuples:

TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
PROPER(f(a)) → c4(F(ok(a)))

Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))

K tuples:

TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
PROPER(f(a)) → c4(F(ok(a)))

Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

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

(23) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

PROPER(g(a)) → c6(G(ok(a)))

We considered the (Usable) Rules:

g(ok(z0)) → ok(g(z0))
active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
proper(g(z0)) → g(proper(z0))
f(ok(z0)) → ok(f(z0))
f(mark(z0)) → mark(f(z0))
proper(a) → ok(a)
proper(f(z0)) → f(proper(z0))

And the Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))

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(PROPER(x₁)) = [2]   
POL(TOP(x₁)) = [2]x₁   
POL(a) = [2]   
POL(active(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(f(x₁)) = [4]x₁   
POL(g(x₁)) = 0   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = x₁   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))

K tuples:

TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(a)) → c6(G(ok(a)))

Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

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

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

Use narrowing to replace TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))

K tuples:

TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(a)) → c6(G(ok(a)))

Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(a)) → c8(TOP(ok(a)))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))

K tuples:

TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(a)) → c6(G(ok(a)))

Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

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

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

Use narrowing to replace TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(f(z0))) → c9(TOP(f(active(z0))), ACTIVE(f(z0)))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(f(z0))) → c9(TOP(f(active(z0))), ACTIVE(f(z0)))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(f(z0))) → c9(TOP(f(active(z0))), ACTIVE(f(z0)))

K tuples:

TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(a)) → c6(G(ok(a)))

Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

F, G, ACTIVE, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

TOP(mark(a)) → c8(TOP(ok(a)))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(f(z0))) → c9(TOP(f(active(z0))), ACTIVE(f(z0)))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(f(z0))) → c9(TOP(f(active(z0))), ACTIVE(f(z0)))

K tuples:

ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(a)) → c6(G(ok(a)))

Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

F, G, ACTIVE, PROPER, TOP

Compound Symbols:

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

(33) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(a)) → c4(F(ok(a)))
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(a)) → c6(G(ok(a)))
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a))))
TOP(ok(f(z0))) → c9(TOP(f(active(z0))), ACTIVE(f(z0)))

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))

K tuples:none
Defined Rule Symbols:

active, f, proper, g

Defined Pair Symbols:

F, G

Compound Symbols:

c2, c3, c7

(35) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

active(f(f(a))) → mark(f(g(f(a))))
active(f(z0)) → f(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(a) → ok(a)
proper(g(z0)) → g(proper(z0))
g(ok(z0)) → ok(g(z0))

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))

S tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

F, G

Compound Symbols:

c2, c3, c7

(37) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))

We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))

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

POL(F(x₁)) = [4]x₁   
POL(G(x₁)) = [4]x₁   
POL(c2(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(mark(x₁)) = [2] + x₁   
POL(ok(x₁)) = [1] + x₁   

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))

S tuples:none
K tuples:

F(mark(z0)) → c2(F(z0))
F(ok(z0)) → c3(F(z0))
G(ok(z0)) → c7(G(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

F, G

Compound Symbols:

c2, c3, c7

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

The set S is empty

(40) BOUNDS(O(1), O(1))