### (0) Obligation:

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

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

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(a, z0)) → c1(F(b, z0))
ACTIVE(f(z0, z0)) → c2(H(a))
ACTIVE(a) → c3
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(a) → c15
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(b) → c17
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(a, z0)) → c1(F(b, z0))
ACTIVE(f(z0, z0)) → c2(H(a))
ACTIVE(a) → c3
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(a) → c15
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(b) → c17
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

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

Removed 5 trailing nodes:

PROPER(b) → c17
PROPER(a) → c15
ACTIVE(g(a, z0)) → c1(F(b, z0))
ACTIVE(f(z0, z0)) → c2(H(a))
ACTIVE(a) → c3

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19

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

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19

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

Use narrowing to replace ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0)) by

ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(a)) → c4(H(mark(b)), ACTIVE(a))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(a)) → c4(H(mark(b)), ACTIVE(a))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(a)) → c4(H(mark(b)), ACTIVE(a))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4

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

Removed 1 trailing tuple parts

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c4

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

Use narrowing to replace ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) by

ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1), ACTIVE(a))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1), ACTIVE(a))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1), ACTIVE(a))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c4, c5

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

Removed 1 trailing tuple parts

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c4, c5, c5

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

Split RHS of tuples not part of any SCC

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c4, c5, c5, c1

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

Use narrowing to replace ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) by

ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1), ACTIVE(a))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1), ACTIVE(a))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1), ACTIVE(a))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c4, c5, c5, c1, c6

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

Removed 1 trailing tuple parts

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c4, c5, c5, c1, c6, c6

### (21) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c4, c5, c5, c1, c6, c6, c2

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

Use narrowing to replace PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) by

PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)), PROPER(a))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(b)) → c13(H(ok(b)), PROPER(b))

### (24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)), PROPER(a))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(b)) → c13(H(ok(b)), PROPER(b))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)), PROPER(a))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(b)) → c13(H(ok(b)), PROPER(b))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c14, c16, c18, c19, c4, c4, c5, c5, c1, c6, c6, c2, c13

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

Removed 2 trailing tuple parts

### (26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c14, c16, c18, c19, c4, c4, c5, c5, c1, c6, c6, c2, c13, c13

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

Use narrowing to replace PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(b), PROPER(x1))

### (28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(b), PROPER(x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(b), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c16, c18, c19, c4, c4, c5, c5, c1, c6, c6, c2, c13, c13, c14

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

Removed 4 trailing tuple parts

### (30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c16, c18, c19, c4, c4, c5, c5, c1, c6, c6, c2, c13, c13, c14, c14

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

Use narrowing to replace PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))

### (32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c18, c19, c4, c4, c5, c5, c1, c6, c6, c2, c13, c13, c14, c14, c16

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

Removed 4 trailing tuple parts

### (34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c18, c19, c4, c4, c5, c5, c1, c6, c6, c2, c13, c13, c14, c14, c16, c16

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

Use narrowing to replace TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)), PROPER(a))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(b)) → c18(TOP(ok(b)), PROPER(b))

### (36) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)), PROPER(a))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(b)) → c18(TOP(ok(b)), PROPER(b))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)), PROPER(a))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(b)) → c18(TOP(ok(b)), PROPER(b))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c19, c4, c4, c5, c5, c1, c6, c6, c2, c13, c13, c14, c14, c16, c16, c18

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

Removed 2 trailing tuple parts

### (38) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c19, c4, c4, c5, c5, c1, c6, c6, c2, c13, c13, c14, c14, c16, c16, c18, c18

### (39) 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)) → c18(TOP(ok(b)))
We considered the (Usable) Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(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(G(x1, x2)) = 0
POL(H(x1)) = 0
POL(PROPER(x1)) = 0
POL(TOP(x1)) = [2]x1
POL(a) = [1]
POL(active(x1)) = x1
POL(b) = 0
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c13(x1, x2)) = x1 + x2
POL(c14(x1, x2)) = x1 + x2
POL(c14(x1, x2, x3)) = x1 + x2 + x3
POL(c16(x1, x2)) = x1 + x2
POL(c16(x1, x2, x3)) = x1 + x2 + x3
POL(c18(x1)) = x1
POL(c18(x1, x2)) = x1 + x2
POL(c19(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c4(x1)) = x1
POL(c4(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c5(x1, x2)) = x1 + x2
POL(c6(x1)) = x1
POL(c6(x1, x2)) = x1 + x2
POL(c7(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(f(x1, x2)) = [1]
POL(g(x1, x2)) = [1]
POL(h(x1)) = [1]
POL(mark(x1)) = [1]
POL(ok(x1)) = x1
POL(proper(x1)) = 0

### (40) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
K tuples:

TOP(mark(b)) → c18(TOP(ok(b)))
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c19, c4, c4, c5, c5, c1, c6, c6, c2, c13, c13, c14, c14, c16, c16, c18, c18

### (41) 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)) → c18(TOP(ok(a)))
We considered the (Usable) Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(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(G(x1, x2)) = 0
POL(H(x1)) = 0
POL(PROPER(x1)) = 0
POL(TOP(x1)) = x1
POL(a) = [1]
POL(active(x1)) = 0
POL(b) = 0
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c13(x1, x2)) = x1 + x2
POL(c14(x1, x2)) = x1 + x2
POL(c14(x1, x2, x3)) = x1 + x2 + x3
POL(c16(x1, x2)) = x1 + x2
POL(c16(x1, x2, x3)) = x1 + x2 + x3
POL(c18(x1)) = x1
POL(c18(x1, x2)) = x1 + x2
POL(c19(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c4(x1)) = x1
POL(c4(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c5(x1, x2)) = x1 + x2
POL(c6(x1)) = x1
POL(c6(x1, x2)) = x1 + x2
POL(c7(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(f(x1, x2)) = 0
POL(g(x1, x2)) = 0
POL(h(x1)) = 0
POL(mark(x1)) = x1
POL(ok(x1)) = 0
POL(proper(x1)) = 0

### (42) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
K tuples:

TOP(mark(b)) → c18(TOP(ok(b)))
TOP(mark(a)) → c18(TOP(ok(a)))
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c19, c4, c4, c5, c5, c1, c6, c6, c2, c13, c13, c14, c14, c16, c16, c18, c18

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

Use narrowing to replace TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(h(z0))) → c19(TOP(mark(g(z0, z0))), ACTIVE(h(z0)))
TOP(ok(g(a, z0))) → c19(TOP(mark(f(b, z0))), ACTIVE(g(a, z0)))
TOP(ok(f(z0, z0))) → c19(TOP(mark(h(a))), ACTIVE(f(z0, z0)))
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a))
TOP(ok(h(z0))) → c19(TOP(h(active(z0))), ACTIVE(h(z0)))
TOP(ok(g(z0, z1))) → c19(TOP(g(active(z0), z1)), ACTIVE(g(z0, z1)))
TOP(ok(f(z0, z1))) → c19(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))

### (44) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))
TOP(ok(h(z0))) → c19(TOP(mark(g(z0, z0))), ACTIVE(h(z0)))
TOP(ok(g(a, z0))) → c19(TOP(mark(f(b, z0))), ACTIVE(g(a, z0)))
TOP(ok(f(z0, z0))) → c19(TOP(mark(h(a))), ACTIVE(f(z0, z0)))
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a))
TOP(ok(h(z0))) → c19(TOP(h(active(z0))), ACTIVE(h(z0)))
TOP(ok(g(z0, z1))) → c19(TOP(g(active(z0), z1)), ACTIVE(g(z0, z1)))
TOP(ok(f(z0, z1))) → c19(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(ok(h(z0))) → c19(TOP(mark(g(z0, z0))), ACTIVE(h(z0)))
TOP(ok(g(a, z0))) → c19(TOP(mark(f(b, z0))), ACTIVE(g(a, z0)))
TOP(ok(f(z0, z0))) → c19(TOP(mark(h(a))), ACTIVE(f(z0, z0)))
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a))
TOP(ok(h(z0))) → c19(TOP(h(active(z0))), ACTIVE(h(z0)))
TOP(ok(g(z0, z1))) → c19(TOP(g(active(z0), z1)), ACTIVE(g(z0, z1)))
TOP(ok(f(z0, z1))) → c19(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
K tuples:

TOP(mark(b)) → c18(TOP(ok(b)))
TOP(mark(a)) → c18(TOP(ok(a)))
Defined Rule Symbols:

active, h, g, f, proper

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c4, c4, c5, c5, c1, c6, c6, c2, c13, c13, c14, c14, c16, c16, c18, c18, c19

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

Removed 3 trailing nodes:

TOP(mark(a)) → c18(TOP(ok(a)))
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a))
TOP(mark(b)) → c18(TOP(ok(b)))

### (46) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c4(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
ACTIVE(h(g(a, z0))) → c1(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c1(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c1(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c1(ACTIVE(g(a, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
ACTIVE(f(g(a, z0), x1)) → c2(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)))
PROPER(h(b)) → c13(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f