YES(O(1), O(n^1)) 75.86/25.39 YES(O(1), O(n^1)) 75.86/25.41 75.86/25.41 75.86/25.41 75.86/25.41 75.86/25.41 75.86/25.41 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 75.86/25.41 75.86/25.41 75.86/25.41
75.86/25.41 76.11/25.41 76.11/25.41
76.11/25.41
76.11/25.41

(0) Obligation:

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

active(f(b, X, c)) → mark(f(X, c, X)) 76.11/25.41
active(c) → mark(b) 76.11/25.41
active(f(X1, X2, X3)) → f(X1, active(X2), X3) 76.11/25.41
f(X1, mark(X2), X3) → mark(f(X1, X2, X3)) 76.11/25.41
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3)) 76.11/25.41
proper(b) → ok(b) 76.11/25.41
proper(c) → ok(c) 76.11/25.41
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3)) 76.11/25.41
top(mark(X)) → top(proper(X)) 76.11/25.41
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
76.11/25.41
76.11/25.41

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

Converted CpxTRS to CDT
76.11/25.41
76.11/25.41

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.11/25.41
active(c) → mark(b) 76.11/25.41
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.11/25.41
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.11/25.41
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.11/25.41
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.11/25.41
proper(b) → ok(b) 76.11/25.41
proper(c) → ok(c) 76.11/25.41
top(mark(z0)) → top(proper(z0)) 76.11/25.41
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(b, z0, c)) → c1(F(z0, c, z0)) 76.11/25.41
ACTIVE(f(z0, z1, z2)) → c3(F(z0, active(z1), z2), ACTIVE(z1)) 76.11/25.41
F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.41
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.41
PROPER(f(z0, z1, z2)) → c6(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 76.11/25.41
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.41
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(b, z0, c)) → c1(F(z0, c, z0)) 76.11/25.41
ACTIVE(f(z0, z1, z2)) → c3(F(z0, active(z1), z2), ACTIVE(z1)) 76.11/25.41
F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.41
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.41
PROPER(f(z0, z1, z2)) → c6(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 76.11/25.41
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.41
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c3, c4, c5, c6, c9, c10

76.11/25.41
76.11/25.41

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

Removed 1 trailing tuple parts
76.11/25.41
76.11/25.41

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.11/25.41
active(c) → mark(b) 76.11/25.41
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.11/25.41
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.11/25.41
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.11/25.41
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.11/25.41
proper(b) → ok(b) 76.11/25.41
proper(c) → ok(c) 76.11/25.41
top(mark(z0)) → top(proper(z0)) 76.11/25.41
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, z1, z2)) → c3(F(z0, active(z1), z2), ACTIVE(z1)) 76.11/25.41
F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.41
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.41
PROPER(f(z0, z1, z2)) → c6(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 76.11/25.41
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.41
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.41
ACTIVE(f(b, z0, c)) → c1
S tuples:

ACTIVE(f(z0, z1, z2)) → c3(F(z0, active(z1), z2), ACTIVE(z1)) 76.11/25.41
F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.41
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.45
PROPER(f(z0, z1, z2)) → c6(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 76.11/25.45
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.45
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.45
ACTIVE(f(b, z0, c)) → c1
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c3, c4, c5, c6, c9, c10, c1

76.11/25.45
76.11/25.45

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

Removed 1 trailing nodes:

ACTIVE(f(b, z0, c)) → c1
76.11/25.45
76.11/25.45

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.11/25.45
active(c) → mark(b) 76.11/25.45
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.11/25.45
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.11/25.45
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.11/25.45
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.11/25.45
proper(b) → ok(b) 76.11/25.45
proper(c) → ok(c) 76.11/25.45
top(mark(z0)) → top(proper(z0)) 76.11/25.45
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, z1, z2)) → c3(F(z0, active(z1), z2), ACTIVE(z1)) 76.11/25.45
F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.45
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.45
PROPER(f(z0, z1, z2)) → c6(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 76.11/25.45
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.45
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.45
ACTIVE(f(b, z0, c)) → c1
S tuples:

ACTIVE(f(z0, z1, z2)) → c3(F(z0, active(z1), z2), ACTIVE(z1)) 76.11/25.45
F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.45
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.45
PROPER(f(z0, z1, z2)) → c6(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 76.11/25.45
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.45
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.45
ACTIVE(f(b, z0, c)) → c1
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c3, c4, c5, c6, c9, c10, c1

76.11/25.45
76.11/25.45

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

Use narrowing to replace ACTIVE(f(z0, z1, z2)) → c3(F(z0, active(z1), z2), ACTIVE(z1)) by

ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.11/25.45
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2), ACTIVE(c)) 76.11/25.45
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2)))
76.11/25.45
76.11/25.45

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.11/25.45
active(c) → mark(b) 76.11/25.45
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.11/25.45
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.11/25.45
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.11/25.45
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.11/25.45
proper(b) → ok(b) 76.11/25.45
proper(c) → ok(c) 76.11/25.45
top(mark(z0)) → top(proper(z0)) 76.11/25.45
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.45
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.45
PROPER(f(z0, z1, z2)) → c6(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 76.11/25.45
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.45
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.45
ACTIVE(f(b, z0, c)) → c1 76.11/25.45
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.11/25.45
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2), ACTIVE(c)) 76.11/25.45
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2)))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.45
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.45
PROPER(f(z0, z1, z2)) → c6(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 76.11/25.45
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.45
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.45
ACTIVE(f(b, z0, c)) → c1 76.11/25.45
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.11/25.45
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2), ACTIVE(c)) 76.11/25.45
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2)))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c4, c5, c6, c9, c10, c1, c3

76.11/25.45
76.11/25.45

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

Removed 1 trailing tuple parts
76.11/25.45
76.11/25.45

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.11/25.45
active(c) → mark(b) 76.11/25.45
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.11/25.45
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.11/25.45
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.11/25.45
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.11/25.45
proper(b) → ok(b) 76.11/25.45
proper(c) → ok(c) 76.11/25.45
top(mark(z0)) → top(proper(z0)) 76.11/25.45
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.45
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.45
PROPER(f(z0, z1, z2)) → c6(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 76.11/25.45
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.45
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.45
ACTIVE(f(b, z0, c)) → c1 76.11/25.45
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.11/25.45
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.11/25.45
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.45
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.45
PROPER(f(z0, z1, z2)) → c6(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 76.11/25.45
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.45
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.45
ACTIVE(f(b, z0, c)) → c1 76.11/25.45
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.11/25.45
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.11/25.45
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c4, c5, c6, c9, c10, c1, c3, c3

76.11/25.45
76.11/25.45

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

Removed 1 trailing nodes:

ACTIVE(f(b, z0, c)) → c1
76.11/25.45
76.11/25.45

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.11/25.45
active(c) → mark(b) 76.11/25.45
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.11/25.45
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.11/25.45
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.11/25.45
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.11/25.45
proper(b) → ok(b) 76.11/25.45
proper(c) → ok(c) 76.11/25.45
top(mark(z0)) → top(proper(z0)) 76.11/25.45
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.45
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.45
PROPER(f(z0, z1, z2)) → c6(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 76.11/25.45
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.45
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.45
ACTIVE(f(b, z0, c)) → c1 76.11/25.45
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.11/25.45
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.11/25.45
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.45
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.45
PROPER(f(z0, z1, z2)) → c6(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 76.11/25.45
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.45
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.45
ACTIVE(f(b, z0, c)) → c1 76.11/25.45
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.11/25.45
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.11/25.46
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c4, c5, c6, c9, c10, c1, c3, c3

76.11/25.46
76.11/25.46

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

Use narrowing to replace PROPER(f(z0, z1, z2)) → c6(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.11/25.46
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 76.11/25.46
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 76.11/25.46
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.11/25.46
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 76.11/25.46
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 76.11/25.46
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.11/25.46
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 76.11/25.46
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
76.11/25.46
76.11/25.46

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.11/25.46
active(c) → mark(b) 76.11/25.46
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.11/25.46
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.11/25.46
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.11/25.46
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.11/25.46
proper(b) → ok(b) 76.11/25.46
proper(c) → ok(c) 76.11/25.46
top(mark(z0)) → top(proper(z0)) 76.11/25.46
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.46
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.46
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.46
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.46
ACTIVE(f(b, z0, c)) → c1 76.11/25.46
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.11/25.46
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.11/25.46
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.11/25.46
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.11/25.46
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 76.11/25.46
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 76.11/25.46
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.11/25.46
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 76.11/25.46
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 76.11/25.46
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.11/25.46
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 76.11/25.46
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.46
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.46
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.46
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.46
ACTIVE(f(b, z0, c)) → c1 76.11/25.46
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.11/25.46
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.11/25.46
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.11/25.46
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.11/25.46
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 76.11/25.46
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 76.11/25.46
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.11/25.46
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 76.11/25.46
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 76.11/25.46
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.11/25.46
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 76.11/25.46
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c4, c5, c9, c10, c1, c3, c3, c6

76.11/25.46
76.11/25.46

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

Removed 6 trailing tuple parts
76.11/25.46
76.11/25.46

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.11/25.46
active(c) → mark(b) 76.11/25.46
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.11/25.46
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.11/25.46
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.11/25.46
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.11/25.46
proper(b) → ok(b) 76.11/25.46
proper(c) → ok(c) 76.11/25.46
top(mark(z0)) → top(proper(z0)) 76.11/25.46
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.46
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.46
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.46
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.46
ACTIVE(f(b, z0, c)) → c1 76.11/25.46
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.11/25.46
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.11/25.46
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.11/25.46
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.11/25.46
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.11/25.46
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.11/25.46
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.11/25.46
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.11/25.46
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.11/25.46
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.11/25.46
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.11/25.46
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.11/25.46
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.11/25.46
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.11/25.46
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.11/25.46
ACTIVE(f(b, z0, c)) → c1 76.11/25.46
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.50
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.50
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.50
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.50
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.50
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.50
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.50
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.50
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.50
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.50
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.50
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c4, c5, c9, c10, c1, c3, c3, c6, c6

76.46/25.50
76.46/25.50

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

Removed 1 trailing nodes:

ACTIVE(f(b, z0, c)) → c1
76.46/25.50
76.46/25.50

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.50
active(c) → mark(b) 76.46/25.50
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.46/25.50
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.50
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.50
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.50
proper(b) → ok(b) 76.46/25.50
proper(c) → ok(c) 76.46/25.50
top(mark(z0)) → top(proper(z0)) 76.46/25.50
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.50
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.50
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.46/25.50
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.50
ACTIVE(f(b, z0, c)) → c1 76.46/25.50
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.50
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.50
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.50
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.50
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.50
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.50
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.50
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.50
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.50
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.50
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.50
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.50
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.50
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) 76.46/25.50
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.50
ACTIVE(f(b, z0, c)) → c1 76.46/25.50
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.50
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.50
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.50
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.50
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.50
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.50
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.50
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.50
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.50
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.50
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.50
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c4, c5, c9, c10, c1, c3, c3, c6, c6

76.46/25.50
76.46/25.50

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

Use narrowing to replace TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.50
TOP(mark(b)) → c9(TOP(ok(b)), PROPER(b)) 76.46/25.50
TOP(mark(c)) → c9(TOP(ok(c)), PROPER(c))
76.46/25.50
76.46/25.50

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.50
active(c) → mark(b) 76.46/25.50
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.46/25.50
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.50
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.50
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.50
proper(b) → ok(b) 76.46/25.50
proper(c) → ok(c) 76.46/25.50
top(mark(z0)) → top(proper(z0)) 76.46/25.50
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.50
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.50
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.50
ACTIVE(f(b, z0, c)) → c1 76.46/25.50
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.50
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.50
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.50
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.50
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.50
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.50
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.50
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.50
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.50
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.50
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.50
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.50
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.50
TOP(mark(b)) → c9(TOP(ok(b)), PROPER(b)) 76.46/25.50
TOP(mark(c)) → c9(TOP(ok(c)), PROPER(c))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.52
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.52
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.52
ACTIVE(f(b, z0, c)) → c1 76.46/25.52
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.52
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.52
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.52
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.52
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.52
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.52
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.52
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.52
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.52
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.52
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.52
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.52
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.52
TOP(mark(b)) → c9(TOP(ok(b)), PROPER(b)) 76.46/25.52
TOP(mark(c)) → c9(TOP(ok(c)), PROPER(c))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c4, c5, c10, c1, c3, c3, c6, c6, c9

76.46/25.52
76.46/25.52

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

Removed 2 trailing tuple parts
76.46/25.52
76.46/25.52

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.52
active(c) → mark(b) 76.46/25.52
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.46/25.52
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.52
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.52
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.52
proper(b) → ok(b) 76.46/25.52
proper(c) → ok(c) 76.46/25.52
top(mark(z0)) → top(proper(z0)) 76.46/25.52
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.52
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.52
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.52
ACTIVE(f(b, z0, c)) → c1 76.46/25.52
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.52
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.52
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.52
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.52
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.52
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.52
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.52
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.52
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.52
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.52
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.52
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.52
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.52
TOP(mark(b)) → c9(TOP(ok(b))) 76.46/25.52
TOP(mark(c)) → c9(TOP(ok(c)))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.52
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.52
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.52
ACTIVE(f(b, z0, c)) → c1 76.46/25.52
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.52
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.52
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.52
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.52
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.52
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.52
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.52
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.52
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.52
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.52
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.52
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.52
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.52
TOP(mark(b)) → c9(TOP(ok(b))) 76.46/25.52
TOP(mark(c)) → c9(TOP(ok(c)))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c4, c5, c10, c1, c3, c3, c6, c6, c9, c9

76.46/25.52
76.46/25.52

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

Removed 1 trailing nodes:

ACTIVE(f(b, z0, c)) → c1
76.46/25.52
76.46/25.52

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.52
active(c) → mark(b) 76.46/25.52
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.46/25.52
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.52
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.52
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.52
proper(b) → ok(b) 76.46/25.52
proper(c) → ok(c) 76.46/25.52
top(mark(z0)) → top(proper(z0)) 76.46/25.52
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.52
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.52
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.52
ACTIVE(f(b, z0, c)) → c1 76.46/25.52
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.52
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.52
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.52
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.52
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.52
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.52
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.52
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.52
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.52
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.52
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.52
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.52
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.52
TOP(mark(b)) → c9(TOP(ok(b))) 76.46/25.52
TOP(mark(c)) → c9(TOP(ok(c)))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.52
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.52
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.52
ACTIVE(f(b, z0, c)) → c1 76.46/25.52
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.52
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.52
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.52
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.52
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.52
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.52
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.52
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.52
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.52
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.52
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.52
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.52
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.52
TOP(mark(b)) → c9(TOP(ok(b))) 76.46/25.52
TOP(mark(c)) → c9(TOP(ok(c)))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c4, c5, c10, c1, c3, c3, c6, c6, c9, c9

76.46/25.52
76.46/25.52

(25) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

TOP(mark(c)) → c9(TOP(ok(c)))
We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.52
proper(b) → ok(b) 76.46/25.52
proper(c) → ok(c) 76.46/25.52
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.52
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.52
active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.52
active(c) → mark(b) 76.46/25.53
active(f(z0, z1, z2)) → f(z0, active(z1), z2)
And the Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.53
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.53
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.53
ACTIVE(f(b, z0, c)) → c1 76.46/25.53
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.53
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.53
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.53
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.53
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.53
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.53
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.53
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.53
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.53
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.53
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.53
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.53
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.53
TOP(mark(b)) → c9(TOP(ok(b))) 76.46/25.53
TOP(mark(c)) → c9(TOP(ok(c)))
The order we found is given by the following interpretation:
Polynomial interpretation : 76.46/25.53

POL(ACTIVE(x1)) = 0    76.46/25.53
POL(F(x1, x2, x3)) = 0    76.46/25.53
POL(PROPER(x1)) = 0    76.46/25.53
POL(TOP(x1)) = [4]x1    76.46/25.53
POL(active(x1)) = 0    76.46/25.53
POL(b) = 0    76.46/25.53
POL(c) = [4]    76.46/25.53
POL(c1) = 0    76.46/25.53
POL(c10(x1, x2)) = x1 + x2    76.46/25.53
POL(c3(x1)) = x1    76.46/25.53
POL(c3(x1, x2)) = x1 + x2    76.46/25.53
POL(c4(x1)) = x1    76.46/25.53
POL(c5(x1)) = x1    76.46/25.53
POL(c6(x1, x2, x3)) = x1 + x2 + x3    76.46/25.53
POL(c6(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    76.46/25.53
POL(c9(x1)) = x1    76.46/25.53
POL(c9(x1, x2)) = x1 + x2    76.46/25.53
POL(f(x1, x2, x3)) = 0    76.46/25.53
POL(mark(x1)) = x1    76.46/25.53
POL(ok(x1)) = 0    76.46/25.53
POL(proper(x1)) = 0   
76.46/25.53
76.46/25.53

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.53
active(c) → mark(b) 76.46/25.53
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.46/25.53
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.53
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.53
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.53
proper(b) → ok(b) 76.46/25.53
proper(c) → ok(c) 76.46/25.53
top(mark(z0)) → top(proper(z0)) 76.46/25.53
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.53
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.53
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.53
ACTIVE(f(b, z0, c)) → c1 76.46/25.53
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.53
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.53
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.53
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.53
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.53
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.53
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.53
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.53
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.53
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.53
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.53
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.53
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.53
TOP(mark(b)) → c9(TOP(ok(b))) 76.46/25.53
TOP(mark(c)) → c9(TOP(ok(c)))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.53
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.53
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.53
ACTIVE(f(b, z0, c)) → c1 76.46/25.53
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.53
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.53
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.53
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.53
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.53
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.53
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.53
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.53
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.53
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.53
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.53
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.53
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.53
TOP(mark(b)) → c9(TOP(ok(b)))
K tuples:

TOP(mark(c)) → c9(TOP(ok(c)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c4, c5, c10, c1, c3, c3, c6, c6, c9, c9

76.46/25.53
76.46/25.53

(27) CdtPolyRedPairProof (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)) → c9(TOP(ok(b)))
We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.53
proper(b) → ok(b) 76.46/25.53
proper(c) → ok(c) 76.46/25.53
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.53
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.53
active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.53
active(c) → mark(b) 76.46/25.53
active(f(z0, z1, z2)) → f(z0, active(z1), z2)
And the Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.53
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.53
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.53
ACTIVE(f(b, z0, c)) → c1 76.46/25.53
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.53
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.53
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.53
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.53
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.53
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.53
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.53
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.53
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.53
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.53
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.53
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.53
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.53
TOP(mark(b)) → c9(TOP(ok(b))) 76.46/25.53
TOP(mark(c)) → c9(TOP(ok(c)))
The order we found is given by the following interpretation:
Polynomial interpretation : 76.46/25.53

POL(ACTIVE(x1)) = 0    76.46/25.53
POL(F(x1, x2, x3)) = 0    76.46/25.53
POL(PROPER(x1)) = 0    76.46/25.53
POL(TOP(x1)) = [4]x1    76.46/25.53
POL(active(x1)) = x1    76.46/25.53
POL(b) = 0    76.46/25.53
POL(c) = [1]    76.46/25.53
POL(c1) = 0    76.46/25.53
POL(c10(x1, x2)) = x1 + x2    76.46/25.53
POL(c3(x1)) = x1    76.46/25.53
POL(c3(x1, x2)) = x1 + x2    76.46/25.53
POL(c4(x1)) = x1    76.46/25.53
POL(c5(x1)) = x1    76.46/25.53
POL(c6(x1, x2, x3)) = x1 + x2 + x3    76.46/25.53
POL(c6(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    76.46/25.53
POL(c9(x1)) = x1    76.46/25.53
POL(c9(x1, x2)) = x1 + x2    76.46/25.53
POL(f(x1, x2, x3)) = [1]    76.46/25.53
POL(mark(x1)) = [1]    76.46/25.53
POL(ok(x1)) = x1    76.46/25.53
POL(proper(x1)) = 0   
76.46/25.53
76.46/25.53

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.53
active(c) → mark(b) 76.46/25.53
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.46/25.53
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.53
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.53
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.53
proper(b) → ok(b) 76.46/25.53
proper(c) → ok(c) 76.46/25.53
top(mark(z0)) → top(proper(z0)) 76.46/25.53
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.53
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.53
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.53
ACTIVE(f(b, z0, c)) → c1 76.46/25.53
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.53
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.53
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.53
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.53
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.53
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.53
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.54
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.54
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.54
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.54
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.54
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.54
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.54
TOP(mark(b)) → c9(TOP(ok(b))) 76.46/25.54
TOP(mark(c)) → c9(TOP(ok(c)))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.54
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) 76.46/25.54
ACTIVE(f(b, z0, c)) → c1 76.46/25.54
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.54
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.54
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.54
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.54
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.54
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.54
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.54
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.54
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.54
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.54
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.54
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.54
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
K tuples:

TOP(mark(c)) → c9(TOP(ok(c))) 76.46/25.54
TOP(mark(b)) → c9(TOP(ok(b)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c4, c5, c10, c1, c3, c3, c6, c6, c9, c9

76.46/25.54
76.46/25.54

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

Use narrowing to replace TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(b, z0, c))) → c10(TOP(mark(f(z0, c, z0))), ACTIVE(f(b, z0, c))) 76.46/25.54
TOP(ok(c)) → c10(TOP(mark(b)), ACTIVE(c)) 76.46/25.54
TOP(ok(f(z0, z1, z2))) → c10(TOP(f(z0, active(z1), z2)), ACTIVE(f(z0, z1, z2)))
76.46/25.54
76.46/25.54

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.54
active(c) → mark(b) 76.46/25.54
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.46/25.54
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.54
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.54
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.54
proper(b) → ok(b) 76.46/25.54
proper(c) → ok(c) 76.46/25.54
top(mark(z0)) → top(proper(z0)) 76.46/25.54
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.54
ACTIVE(f(b, z0, c)) → c1 76.46/25.54
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.54
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.54
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.54
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.54
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.54
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.54
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.54
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.54
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.54
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.54
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.54
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.54
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.54
TOP(mark(b)) → c9(TOP(ok(b))) 76.46/25.54
TOP(mark(c)) → c9(TOP(ok(c))) 76.46/25.54
TOP(ok(f(b, z0, c))) → c10(TOP(mark(f(z0, c, z0))), ACTIVE(f(b, z0, c))) 76.46/25.54
TOP(ok(c)) → c10(TOP(mark(b)), ACTIVE(c)) 76.46/25.54
TOP(ok(f(z0, z1, z2))) → c10(TOP(f(z0, active(z1), z2)), ACTIVE(f(z0, z1, z2)))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.54
ACTIVE(f(b, z0, c)) → c1 76.46/25.54
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.54
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.54
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.54
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.54
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.54
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.54
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.54
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.54
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.54
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.54
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.54
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.54
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.54
TOP(ok(f(b, z0, c))) → c10(TOP(mark(f(z0, c, z0))), ACTIVE(f(b, z0, c))) 76.46/25.54
TOP(ok(c)) → c10(TOP(mark(b)), ACTIVE(c)) 76.46/25.54
TOP(ok(f(z0, z1, z2))) → c10(TOP(f(z0, active(z1), z2)), ACTIVE(f(z0, z1, z2)))
K tuples:

TOP(mark(c)) → c9(TOP(ok(c))) 76.46/25.54
TOP(mark(b)) → c9(TOP(ok(b)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, ACTIVE, PROPER, TOP

Compound Symbols:

c4, c5, c1, c3, c3, c6, c6, c9, c9, c10

76.46/25.54
76.46/25.54

(31) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(f(b, z0, c)) → c1 76.46/25.54
ACTIVE(f(x0, f(b, z0, c), x2)) → c3(F(x0, mark(f(z0, c, z0)), x2), ACTIVE(f(b, z0, c))) 76.46/25.54
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c3(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 76.46/25.54
ACTIVE(f(x0, c, x2)) → c3(F(x0, mark(b), x2)) 76.46/25.54
PROPER(f(x0, x1, f(z0, z1, z2))) → c6(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 76.46/25.54
PROPER(f(x0, f(z0, z1, z2), x2)) → c6(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 76.46/25.54
PROPER(f(f(z0, z1, z2), x1, x2)) → c6(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 76.46/25.54
PROPER(f(x0, x1, b)) → c6(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 76.46/25.54
PROPER(f(x0, x1, c)) → c6(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 76.46/25.54
PROPER(f(x0, b, x2)) → c6(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.54
PROPER(f(x0, c, x2)) → c6(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 76.46/25.54
PROPER(f(b, x1, x2)) → c6(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.54
PROPER(f(c, x1, x2)) → c6(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 76.46/25.54
TOP(mark(f(z0, z1, z2))) → c9(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 76.46/25.54
TOP(ok(f(b, z0, c))) → c10(TOP(mark(f(z0, c, z0))), ACTIVE(f(b, z0, c))) 76.46/25.54
TOP(ok(f(z0, z1, z2))) → c10(TOP(f(z0, active(z1), z2)), ACTIVE(f(z0, z1, z2)))
76.46/25.54
76.46/25.54

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.54
active(c) → mark(b) 76.46/25.54
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.46/25.54
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.54
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.54
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.54
proper(b) → ok(b) 76.46/25.54
proper(c) → ok(c) 76.46/25.54
top(mark(z0)) → top(proper(z0)) 76.46/25.54
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.54
TOP(mark(b)) → c9(TOP(ok(b))) 76.46/25.54
TOP(mark(c)) → c9(TOP(ok(c))) 76.46/25.54
TOP(ok(c)) → c10(TOP(mark(b)), ACTIVE(c))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.54
TOP(ok(c)) → c10(TOP(mark(b)), ACTIVE(c))
K tuples:

TOP(mark(c)) → c9(TOP(ok(c))) 76.46/25.54
TOP(mark(b)) → c9(TOP(ok(b)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP

Compound Symbols:

c4, c5, c9, c10

76.46/25.54
76.46/25.54

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

Removed 2 trailing tuple parts
76.46/25.54
76.46/25.54

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.54
active(c) → mark(b) 76.46/25.54
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.46/25.54
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.54
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.54
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.54
proper(b) → ok(b) 76.46/25.54
proper(c) → ok(c) 76.46/25.54
top(mark(z0)) → top(proper(z0)) 76.46/25.54
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.54
TOP(mark(c)) → c9(TOP(ok(c))) 76.46/25.54
TOP(mark(b)) → c9 76.46/25.54
TOP(ok(c)) → c10(TOP(mark(b)))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2)) 76.46/25.54
TOP(ok(c)) → c10(TOP(mark(b)))
K tuples:

TOP(mark(c)) → c9(TOP(ok(c))) 76.46/25.54
TOP(mark(b)) → c9
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP

Compound Symbols:

c4, c5, c9, c9, c10

76.46/25.54
76.46/25.54

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

Removed 3 trailing nodes:

TOP(mark(b)) → c9 76.46/25.54
TOP(mark(c)) → c9(TOP(ok(c))) 76.46/25.54
TOP(ok(c)) → c10(TOP(mark(b)))
76.46/25.54
76.46/25.54

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.54
active(c) → mark(b) 76.46/25.54
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.46/25.54
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.54
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.54
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.54
proper(b) → ok(b) 76.46/25.54
proper(c) → ok(c) 76.46/25.54
top(mark(z0)) → top(proper(z0)) 76.46/25.54
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2))
S tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c4, c5

76.46/25.54
76.46/25.54

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

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

F(z0, mark(z1), z2) → c4(F(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 76.46/25.54

POL(F(x1, x2, x3)) = [4]x2    76.46/25.54
POL(c4(x1)) = x1    76.46/25.54
POL(c5(x1)) = x1    76.46/25.54
POL(mark(x1)) = [4] + x1    76.46/25.54
POL(ok(x1)) = x1   
76.46/25.54
76.46/25.54

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.54
active(c) → mark(b) 76.46/25.54
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.46/25.54
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.54
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.54
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.54
proper(b) → ok(b) 76.46/25.54
proper(c) → ok(c) 76.46/25.54
top(mark(z0)) → top(proper(z0)) 76.46/25.54
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2))
S tuples:

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

F(z0, mark(z1), z2) → c4(F(z0, z1, z2))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c4, c5

76.46/25.54
76.46/25.54

(39) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

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

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 76.46/25.54

POL(F(x1, x2, x3)) = x3    76.46/25.54
POL(c4(x1)) = x1    76.46/25.54
POL(c5(x1)) = x1    76.46/25.54
POL(mark(x1)) = x1    76.46/25.54
POL(ok(x1)) = [1] + x1   
76.46/25.54
76.46/25.54

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(b, z0, c)) → mark(f(z0, c, z0)) 76.46/25.54
active(c) → mark(b) 76.46/25.54
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 76.46/25.54
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 76.46/25.54
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 76.46/25.54
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 76.46/25.54
proper(b) → ok(b) 76.46/25.54
proper(c) → ok(c) 76.46/25.54
top(mark(z0)) → top(proper(z0)) 76.46/25.54
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2))
S tuples:none
K tuples:

F(z0, mark(z1), z2) → c4(F(z0, z1, z2)) 76.46/25.54
F(ok(z0), ok(z1), ok(z2)) → c5(F(z0, z1, z2))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c4, c5

76.46/25.54
76.46/25.54

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

The set S is empty
76.46/25.54
76.46/25.54

(42) BOUNDS(O(1), O(1))

76.46/25.54
76.46/25.54
76.83/25.63 EOF