### (0) Obligation:

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

active(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(X)) → ok(g(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, PROPER, G, H, TOP

Compound Symbols:

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

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

Removed 4 trailing nodes:

ACTIVE(c) → c2
PROPER(c) → c6
ACTIVE(h(d)) → c3(G(c))
PROPER(d) → c7

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(g(z0)) → c1(H(z0))
PROPER(g(z0)) → c4(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(g(z0)) → c4(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, PROPER, G, H, TOP

Compound Symbols:

c1, c4, c5, c8, c9, c10, c11

### (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:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
Tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(g(z0)) → c4(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(g(z0)) → c4(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

proper, g, h, active

Defined Pair Symbols:

ACTIVE, PROPER, G, H, TOP

Compound Symbols:

c1, c4, c5, c8, c9, c10, c11

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

Use narrowing to replace PROPER(g(z0)) → c4(G(proper(z0)), PROPER(z0)) by

PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(d))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
Tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(d))
S tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(d))
K tuples:none
Defined Rule Symbols:

proper, g, h, active

Defined Pair Symbols:

ACTIVE, PROPER, G, H, TOP

Compound Symbols:

c1, c5, c8, c9, c10, c11, c4

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

Removed 2 trailing tuple parts

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
Tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
S tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
K tuples:none
Defined Rule Symbols:

proper, g, h, active

Defined Pair Symbols:

ACTIVE, PROPER, G, H, TOP

Compound Symbols:

c1, c5, c8, c9, c10, c11, c4, c4

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

Use narrowing to replace PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0)) by

PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)), PROPER(c))
PROPER(h(d)) → c5(H(ok(d)), PROPER(d))

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)), PROPER(c))
PROPER(h(d)) → c5(H(ok(d)), PROPER(d))
S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)), PROPER(c))
PROPER(h(d)) → c5(H(ok(d)), PROPER(d))
K tuples:none
Defined Rule Symbols:

proper, g, h, active

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c10, c11, c4, c4, c5

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

Removed 2 trailing tuple parts

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
K tuples:none
Defined Rule Symbols:

proper, g, h, active

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c10, c11, c4, c4, c5, c5

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

Use narrowing to replace TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)), PROPER(c))
TOP(mark(d)) → c10(TOP(ok(d)), PROPER(d))

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)), PROPER(c))
TOP(mark(d)) → c10(TOP(ok(d)), PROPER(d))
S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)), PROPER(c))
TOP(mark(d)) → c10(TOP(ok(d)), PROPER(d))
K tuples:none
Defined Rule Symbols:

proper, g, h, active

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c11, c4, c4, c5, c5, c10

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

Removed 2 trailing tuple parts

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))
S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))
K tuples:none
Defined Rule Symbols:

proper, g, h, active

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c11, c4, c4, c5, c5, c10, c10

### (19) 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)) → c10(TOP(ok(c)))
We considered the (Usable) Rules:

g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(h(d)) → mark(g(c))
active(c) → mark(d)
And the Tuples:

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

POL(ACTIVE(x1)) = 0
POL(G(x1)) = 0
POL(H(x1)) = 0
POL(PROPER(x1)) = 0
POL(TOP(x1)) = x1
POL(active(x1)) = 0
POL(c) = [1]
POL(c1(x1)) = x1
POL(c10(x1)) = x1
POL(c10(x1, x2)) = x1 + x2
POL(c11(x1, x2)) = x1 + x2
POL(c4(x1)) = x1
POL(c4(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c5(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(d) = 0
POL(g(x1)) = 0
POL(h(x1)) = 0
POL(mark(x1)) = x1
POL(ok(x1)) = 0
POL(proper(x1)) = 0

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))
S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(d)) → c10(TOP(ok(d)))
K tuples:

TOP(mark(c)) → c10(TOP(ok(c)))
Defined Rule Symbols:

proper, g, h, active

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c11, c4, c4, c5, c5, c10, c10

### (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(d)) → c10(TOP(ok(d)))
We considered the (Usable) Rules:

g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(h(d)) → mark(g(c))
active(c) → mark(d)
And the Tuples:

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

POL(ACTIVE(x1)) = 0
POL(G(x1)) = 0
POL(H(x1)) = 0
POL(PROPER(x1)) = 0
POL(TOP(x1)) = [2]x1
POL(active(x1)) = x1
POL(c) = [2]
POL(c1(x1)) = x1
POL(c10(x1)) = x1
POL(c10(x1, x2)) = x1 + x2
POL(c11(x1, x2)) = x1 + x2
POL(c4(x1)) = x1
POL(c4(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c5(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(d) = 0
POL(g(x1)) = [2]
POL(h(x1)) = [2]
POL(mark(x1)) = [2]
POL(ok(x1)) = x1
POL(proper(x1)) = 0

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))
S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
K tuples:

TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))
Defined Rule Symbols:

proper, g, h, active

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c11, c4, c4, c5, c5, c10, c10

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

Use narrowing to replace TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
TOP(ok(c)) → c11(TOP(mark(d)), ACTIVE(c))
TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))

### (24) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
TOP(ok(c)) → c11(TOP(mark(d)), ACTIVE(c))
TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
TOP(ok(c)) → c11(TOP(mark(d)), ACTIVE(c))
TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
K tuples:

TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))
Defined Rule Symbols:

proper, g, h, active

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

c1, c8, c9, c4, c4, c5, c5, c10, c10, c11

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

Removed 3 trailing nodes:

TOP(ok(c)) → c11(TOP(mark(d)), ACTIVE(c))
TOP(mark(d)) → c10(TOP(ok(d)))
TOP(mark(c)) → c10(TOP(ok(c)))

### (26) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
K tuples:none
Defined Rule Symbols:

proper, g, h, active

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

c1, c8, c9, c4, c4, c5, c5, c10, c11

### (27) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(g(z0)) → c1(H(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
PROPER(g(d)) → c4(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)))
PROPER(h(d)) → c5(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))

### (28) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
Tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
S tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
K tuples:none
Defined Rule Symbols:

proper, g, h, active

Defined Pair Symbols:

G, H

Compound Symbols:

c8, c9

### (29) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))

### (30) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
S tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

G, H

Compound Symbols:

c8, c9

### (31) 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.

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
We considered the (Usable) Rules:none
And the Tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(G(x1)) = [4]x1
POL(H(x1)) = [4]x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(ok(x1)) = [4] + x1

### (32) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
S tuples:none
K tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

G, H

Compound Symbols:

c8, c9

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

The set S is empty