### (0) Obligation:

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

active(f(X, X)) → mark(f(a, b))
active(b) → mark(a)
active(f(X1, X2)) → f(active(X1), X2)
f(mark(X1), X2) → mark(f(X1, X2))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(a) → ok(a)
proper(b) → ok(b)
f(ok(X1), ok(X2)) → ok(f(X1, X2))
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(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

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

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

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

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

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

Removed 4 trailing nodes:

ACTIVE(f(z0, z0)) → c(F(a, b))
ACTIVE(b) → c1
PROPER(a) → c6
PROPER(b) → c7

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
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, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c2, c3, c4, c5, 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(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
Tuples:

ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
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

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c2, c3, c4, c5, c8, c9

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

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

ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1), ACTIVE(b))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1), ACTIVE(b))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1), ACTIVE(b))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c8, c9, c2

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

Removed 1 trailing tuple parts

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
K tuples:none
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c8, c9, c2, c2

### (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(b, x1)) → c2(F(mark(a), x1))
We considered the (Usable) Rules:

active(b) → mark(a)
proper(b) → ok(b)
f(ok(z0), ok(z1)) → ok(f(z0, z1))
active(f(z0, z1)) → f(active(z0), z1)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
f(mark(z0), z1) → mark(f(z0, z1))
active(f(z0, z0)) → mark(f(a, b))
And the Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = [4]x1
POL(F(x1, x2)) = 0
POL(PROPER(x1)) = 0
POL(TOP(x1)) = [4]x1
POL(a) = 0
POL(active(x1)) = 0
POL(b) = [4]
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1, x2, x3)) = x1 + x2 + x3
POL(c8(x1, x2)) = x1 + x2
POL(c9(x1, x2)) = x1 + x2
POL(f(x1, x2)) = [4]x1
POL(mark(x1)) = x1
POL(ok(x1)) = x1
POL(proper(x1)) = x1

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c8, c9, c2, c2

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

Use narrowing to replace PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c8, c9, c2, c2, c5

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

Removed 4 trailing tuple parts

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c8, c9, c2, c2, c5, c5

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

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

TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(b)) → c8(TOP(ok(b)), PROPER(b))

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(b)) → c8(TOP(ok(b)), PROPER(b))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(b)) → c8(TOP(ok(b)), PROPER(b))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c9, c2, c2, c5, c5, c8

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

Removed 2 trailing tuple parts

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c9, c2, c2, c5, c5, c8, c8

### (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(b)) → c8(TOP(ok(b)))
We considered the (Usable) Rules:

active(b) → mark(a)
proper(b) → ok(b)
f(ok(z0), ok(z1)) → ok(f(z0, z1))
active(f(z0, z1)) → f(active(z0), z1)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
f(mark(z0), z1) → mark(f(z0, z1))
active(f(z0, z0)) → mark(f(a, b))
And the Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0
POL(F(x1, x2)) = 0
POL(PROPER(x1)) = 0
POL(TOP(x1)) = x1
POL(a) = 0
POL(active(x1)) = 0
POL(b) = [4]
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1, x2)) = x1 + x2
POL(c5(x1, x2, x3)) = x1 + x2 + x3
POL(c8(x1)) = x1
POL(c8(x1, x2)) = x1 + x2
POL(c9(x1, x2)) = x1 + x2
POL(f(x1, x2)) = [4]x1
POL(mark(x1)) = x1
POL(ok(x1)) = 0
POL(proper(x1)) = 0

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
TOP(mark(b)) → c8(TOP(ok(b)))
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c9, c2, c2, c5, c5, c8, c8

### (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(a)) → c8(TOP(ok(a)))
We considered the (Usable) Rules:

active(b) → mark(a)
f(ok(z0), ok(z1)) → ok(f(z0, z1))
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
active(f(z0, z0)) → mark(f(a, b))
And the Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0
POL(F(x1, x2)) = 0
POL(PROPER(x1)) = 0
POL(TOP(x1)) = [4]x1
POL(a) = 0
POL(active(x1)) = x1
POL(b) = [1]
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1, x2)) = x1 + x2
POL(c5(x1, x2, x3)) = x1 + x2 + x3
POL(c8(x1)) = x1
POL(c8(x1, x2)) = x1 + x2
POL(c9(x1, x2)) = x1 + x2
POL(f(x1, x2)) = [1]
POL(mark(x1)) = [1]
POL(ok(x1)) = x1
POL(proper(x1)) = 0

### (24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
TOP(mark(b)) → c8(TOP(ok(b)))
TOP(mark(a)) → c8(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c9, c2, c2, c5, c5, c8, c8

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

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

TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0)))
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))

### (26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0)))
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0)))
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
TOP(mark(b)) → c8(TOP(ok(b)))
TOP(mark(a)) → c8(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

F, ACTIVE, PROPER, TOP

Compound Symbols:

c3, c4, c2, c2, c5, c5, c8, c8, c9

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

Removed 3 trailing nodes:

TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))

### (28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0)))
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0)))
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

F, ACTIVE, PROPER, TOP

Compound Symbols:

c3, c4, c2, c2, c5, c5, c8, c9

### (29) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0)))
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))

### (30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
K tuples:none
Defined Rule Symbols:

active, f, proper

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

### (31) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)

### (32) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

### (33) 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), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2)) = [4]x1 + [4]x2
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(mark(x1)) = [2] + x1
POL(ok(x1)) = [4] + x1

### (34) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
S tuples:none
K tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

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

The set S is empty