### (0) Obligation:

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

active(f(b, c, x)) → mark(f(x, x, x))
active(f(x, y, z)) → f(x, y, active(z))
active(d) → m(b)
f(x, y, mark(z)) → mark(f(x, y, z))
active(d) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(x, y, z)) → f(proper(x), proper(y), proper(z))
f(ok(x), ok(y), ok(z)) → ok(f(x, y, z))
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(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c2(F(z0, z1, active(z2)), ACTIVE(z2))
ACTIVE(d) → c3
ACTIVE(d) → c4
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
PROPER(b) → c7
PROPER(c) → c8
PROPER(d) → c9
PROPER(f(z0, z1, z2)) → c10(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c2(F(z0, z1, active(z2)), ACTIVE(z2))
ACTIVE(d) → c3
ACTIVE(d) → c4
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
PROPER(b) → c7
PROPER(c) → c8
PROPER(d) → c9
PROPER(f(z0, z1, z2)) → c10(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
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, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

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

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

Removed 5 trailing nodes:

ACTIVE(d) → c3
ACTIVE(d) → c4
PROPER(b) → c7
PROPER(c) → c8
PROPER(d) → c9

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c2(F(z0, z1, active(z2)), ACTIVE(z2))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
PROPER(f(z0, z1, z2)) → c10(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c2(F(z0, z1, active(z2)), ACTIVE(z2))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
PROPER(f(z0, z1, z2)) → c10(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
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, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c2, c5, c6, c10, c11, c12

### (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(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c2(F(z0, z1, active(z2)), ACTIVE(z2))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
PROPER(f(z0, z1, z2)) → c10(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(z0, z1, z2)) → c2(F(z0, z1, active(z2)), ACTIVE(z2))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
PROPER(f(z0, z1, z2)) → c10(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
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, proper

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c2, c5, c6, c10, c11, c12

### (7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(f(z0, z1, z2)) → c2(F(z0, z1, active(z2)), ACTIVE(z2)) by

ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, m(b)), ACTIVE(d))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)), ACTIVE(d))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
PROPER(f(z0, z1, z2)) → c10(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, m(b)), ACTIVE(d))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)), ACTIVE(d))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
PROPER(f(z0, z1, z2)) → c10(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, m(b)), ACTIVE(d))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)), ACTIVE(d))
K tuples:none
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c5, c6, c10, c11, c12, c2

### (9) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, m(b)), ACTIVE(d))

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
PROPER(f(z0, z1, z2)) → c10(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)), ACTIVE(d))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
PROPER(f(z0, z1, z2)) → c10(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)), ACTIVE(d))
K tuples:none
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c5, c6, c10, c11, c12, c2

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

Removed 1 trailing tuple parts

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
PROPER(f(z0, z1, z2)) → c10(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
PROPER(f(z0, z1, z2)) → c10(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
K tuples:none
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c5, c6, c10, c11, c12, c2, c2

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

Use narrowing to replace PROPER(f(z0, z1, z2)) → c10(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1), PROPER(d))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(d), PROPER(x2))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(d), PROPER(x1), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1), PROPER(d))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(d), PROPER(x2))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(d), PROPER(x1), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1), PROPER(d))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(d), PROPER(x2))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(d), PROPER(x1), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c12, c2, c2, c10

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

Removed 9 trailing tuple parts

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c12, c2, c2, c10, c10

### (17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(b)) → c11(TOP(ok(b)), PROPER(b))
TOP(mark(c)) → c11(TOP(ok(c)), PROPER(c))
TOP(mark(d)) → c11(TOP(ok(d)), PROPER(d))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(b)) → c11(TOP(ok(b)), PROPER(b))
TOP(mark(c)) → c11(TOP(ok(c)), PROPER(c))
TOP(mark(d)) → c11(TOP(ok(d)), PROPER(d))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(b)) → c11(TOP(ok(b)), PROPER(b))
TOP(mark(c)) → c11(TOP(ok(c)), PROPER(c))
TOP(mark(d)) → c11(TOP(ok(d)), PROPER(d))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
K tuples:none
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c12, c2, c2, c10, c10, c11

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

Removed 3 trailing tuple parts

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(b)) → c11(TOP(ok(b)))
TOP(mark(c)) → c11(TOP(ok(c)))
TOP(mark(d)) → c11(TOP(ok(d)))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(b)) → c11(TOP(ok(b)))
TOP(mark(c)) → c11(TOP(ok(c)))
TOP(mark(d)) → c11(TOP(ok(d)))
K tuples:none
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c12, c2, c2, c10, c10, c11, c11

### (21) 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(c)) → c11(TOP(ok(c)))
We considered the (Usable) Rules:

f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
active(d) → mark(c)
active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(d) → m(b)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
And the Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(b)) → c11(TOP(ok(b)))
TOP(mark(c)) → c11(TOP(ok(c)))
TOP(mark(d)) → c11(TOP(ok(d)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0
POL(F(x1, x2, x3)) = 0
POL(PROPER(x1)) = 0
POL(TOP(x1)) = [4]x1
POL(active(x1)) = x1
POL(b) = [1]
POL(c) = 0
POL(c1(x1)) = x1
POL(c10(x1, x2, x3)) = x1 + x2 + x3
POL(c10(x1, x2, x3, x4)) = x1 + x2 + x3 + x4
POL(c11(x1)) = x1
POL(c11(x1, x2)) = x1 + x2
POL(c12(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(d) = [1]
POL(f(x1, x2, x3)) = [1]
POL(m(x1)) = x1
POL(mark(x1)) = [1]
POL(ok(x1)) = x1
POL(proper(x1)) = 0

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(b)) → c11(TOP(ok(b)))
TOP(mark(c)) → c11(TOP(ok(c)))
TOP(mark(d)) → c11(TOP(ok(d)))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(b)) → c11(TOP(ok(b)))
TOP(mark(d)) → c11(TOP(ok(d)))
K tuples:

TOP(mark(c)) → c11(TOP(ok(c)))
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c12, c2, c2, c10, c10, c11, c11

### (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.

TOP(mark(d)) → c11(TOP(ok(d)))
We considered the (Usable) Rules:

f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
active(d) → mark(c)
active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(d) → m(b)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
And the Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(b)) → c11(TOP(ok(b)))
TOP(mark(c)) → c11(TOP(ok(c)))
TOP(mark(d)) → c11(TOP(ok(d)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0
POL(F(x1, x2, x3)) = 0
POL(PROPER(x1)) = 0
POL(TOP(x1)) = [2]x1
POL(active(x1)) = 0
POL(b) = 0
POL(c) = 0
POL(c1(x1)) = x1
POL(c10(x1, x2, x3)) = x1 + x2 + x3
POL(c10(x1, x2, x3, x4)) = x1 + x2 + x3 + x4
POL(c11(x1)) = x1
POL(c11(x1, x2)) = x1 + x2
POL(c12(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(d) = [1]
POL(f(x1, x2, x3)) = 0
POL(m(x1)) = 0
POL(mark(x1)) = x1
POL(ok(x1)) = 0
POL(proper(x1)) = 0

### (24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(b)) → c11(TOP(ok(b)))
TOP(mark(c)) → c11(TOP(ok(c)))
TOP(mark(d)) → c11(TOP(ok(d)))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(b)) → c11(TOP(ok(b)))
K tuples:

TOP(mark(c)) → c11(TOP(ok(c)))
TOP(mark(d)) → c11(TOP(ok(d)))
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c12, c2, c2, c10, c10, c11, c11

### (25) 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(b)) → c11(TOP(ok(b)))
We considered the (Usable) Rules:

f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
active(d) → mark(c)
active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(d) → m(b)
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
And the Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(b)) → c11(TOP(ok(b)))
TOP(mark(c)) → c11(TOP(ok(c)))
TOP(mark(d)) → c11(TOP(ok(d)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0
POL(F(x1, x2, x3)) = 0
POL(PROPER(x1)) = 0
POL(TOP(x1)) = x1
POL(active(x1)) = 0
POL(b) = [4]
POL(c) = 0
POL(c1(x1)) = x1
POL(c10(x1, x2, x3)) = x1 + x2 + x3
POL(c10(x1, x2, x3, x4)) = x1 + x2 + x3 + x4
POL(c11(x1)) = x1
POL(c11(x1, x2)) = x1 + x2
POL(c12(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(d) = [2]
POL(f(x1, x2, x3)) = 0
POL(m(x1)) = 0
POL(mark(x1)) = x1
POL(ok(x1)) = 0
POL(proper(x1)) = 0

### (26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(b)) → c11(TOP(ok(b)))
TOP(mark(c)) → c11(TOP(ok(c)))
TOP(mark(d)) → c11(TOP(ok(d)))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
K tuples:

TOP(mark(c)) → c11(TOP(ok(c)))
TOP(mark(d)) → c11(TOP(ok(d)))
TOP(mark(b)) → c11(TOP(ok(b)))
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c12, c2, c2, c10, c10, c11, c11

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

Use narrowing to replace TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(b, c, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
TOP(ok(d)) → c12(TOP(m(b)), ACTIVE(d))
TOP(ok(d)) → c12(TOP(mark(c)), ACTIVE(d))

### (28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(b)) → c11(TOP(ok(b)))
TOP(mark(c)) → c11(TOP(ok(c)))
TOP(mark(d)) → c11(TOP(ok(d)))
TOP(ok(f(b, c, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
TOP(ok(d)) → c12(TOP(m(b)), ACTIVE(d))
TOP(ok(d)) → c12(TOP(mark(c)), ACTIVE(d))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(ok(f(b, c, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
TOP(ok(d)) → c12(TOP(m(b)), ACTIVE(d))
TOP(ok(d)) → c12(TOP(mark(c)), ACTIVE(d))
K tuples:

TOP(mark(c)) → c11(TOP(ok(c)))
TOP(mark(d)) → c11(TOP(ok(d)))
TOP(mark(b)) → c11(TOP(ok(b)))
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c5, c6, c2, c2, c10, c10, c11, c11, c12

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

Removed 5 trailing nodes:

TOP(mark(d)) → c11(TOP(ok(d)))
TOP(mark(b)) → c11(TOP(ok(b)))
TOP(ok(d)) → c12(TOP(mark(c)), ACTIVE(d))
TOP(mark(c)) → c11(TOP(ok(c)))
TOP(ok(d)) → c12(TOP(m(b)), ACTIVE(d))

### (30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(ok(f(b, c, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
S tuples:

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(ok(f(b, c, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
K tuples:none
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c5, c6, c2, c2, c10, c10, c11, c12

### (31) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(f(b, c, z0)) → c1(F(z0, z0, z0))
ACTIVE(f(x0, x1, f(b, c, z0))) → c2(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c2(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
ACTIVE(f(x0, x1, d)) → c2(F(x0, x1, mark(c)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c10(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c10(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c10(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c10(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c10(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, d)) → c10(F(proper(x0), proper(x1), ok(d)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c10(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c10(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, d, x2)) → c10(F(proper(x0), ok(d), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c10(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c10(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(d, x1, x2)) → c10(F(ok(d), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(ok(f(b, c, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(b, c, z0)))
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))

### (32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
Tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
S tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

F

Compound Symbols:

c5, c6

### (33) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

active(f(b, c, z0)) → mark(f(z0, z0, z0))
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
active(d) → m(b)
active(d) → mark(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2))
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))

### (34) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
S tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

F

Compound Symbols:

c5, c6

### (35) 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(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2, x3)) = [4]x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(mark(x1)) = [3]
POL(ok(x1)) = [2] + x1

### (36) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
S tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
K tuples:

F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
Defined Rule Symbols:none

Defined Pair Symbols:

F

Compound Symbols:

c5, c6

### (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(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2, x3)) = [4]x1 + [5]x2 + [2]x3
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(mark(x1)) = [1] + x1
POL(ok(x1)) = x1

### (38) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
S tuples:none
K tuples:

F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
Defined Rule Symbols:none

Defined Pair Symbols:

F

Compound Symbols:

c5, c6

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

The set S is empty