YES(O(1), O(n^2)) 60.95/21.52 YES(O(1), O(n^2)) 60.95/21.53 60.95/21.53 60.95/21.53 60.95/21.53 60.95/21.53 60.95/21.53 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 60.95/21.53 60.95/21.53 60.95/21.53
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
60.95/21.53 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
60.95/21.53 
↳2 CdtProblem
60.95/21.53 
↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
60.95/21.53 
↳4 CdtProblem
60.95/21.53 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
60.95/21.53 
↳6 CdtProblem
60.95/21.53 
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
60.95/21.53 
↳8 CdtProblem
60.95/21.53 
↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
60.95/21.53 
↳10 CdtProblem
60.95/21.53 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
60.95/21.53 
↳12 CdtProblem
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↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
60.95/21.53 
↳14 CdtProblem
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↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
60.95/21.53 
↳16 CdtProblem
60.95/21.53 
↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
60.95/21.53 
↳18 CdtProblem
60.95/21.53 
↳19 CdtUnreachableProof (⇔)
60.95/21.53 
↳20 CdtProblem
60.95/21.53 
↳21 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
60.95/21.53 
↳22 CdtProblem
60.95/21.53 
↳23 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
60.95/21.53 
↳24 CdtProblem
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↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳26 CdtProblem
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↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳28 CdtProblem
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↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
60.95/21.53 
↳30 CdtProblem
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↳31 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳32 BOUNDS(O(1), O(1))
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60.95/21.53

(0) Obligation:

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

active(f(a, b, X)) → mark(f(X, X, X)) 60.95/21.53
active(c) → mark(a) 60.95/21.53
active(c) → mark(b) 60.95/21.53
active(f(X1, X2, X3)) → f(active(X1), X2, X3) 60.95/21.53
active(f(X1, X2, X3)) → f(X1, X2, active(X3)) 60.95/21.53
f(mark(X1), X2, X3) → mark(f(X1, X2, X3)) 60.95/21.53
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3)) 60.95/21.53
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3)) 60.95/21.53
proper(a) → ok(a) 60.95/21.53
proper(b) → ok(b) 60.95/21.53
proper(c) → ok(c) 60.95/21.53
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3)) 60.95/21.53
top(mark(X)) → top(proper(X)) 60.95/21.59
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
60.95/21.59
60.95/21.59

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

Converted CpxTRS to CDT
60.95/21.59
60.95/21.59

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 60.95/21.59
active(c) → mark(a) 60.95/21.59
active(c) → mark(b) 60.95/21.59
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 60.95/21.59
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 60.95/21.59
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 60.95/21.59
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 60.95/21.59
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 60.95/21.59
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 60.95/21.59
proper(a) → ok(a) 60.95/21.59
proper(b) → ok(b) 60.95/21.59
proper(c) → ok(c) 60.95/21.59
top(mark(z0)) → top(proper(z0)) 60.95/21.59
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 60.95/21.59
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 60.95/21.59
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 60.95/21.59
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 60.95/21.59
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 60.95/21.59
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 60.95/21.59
PROPER(f(z0, z1, z2)) → c9(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 60.95/21.59
TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0)) 60.95/21.59
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
PROPER(f(z0, z1, z2)) → c9(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 61.31/21.61
TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0)) 61.31/21.61
TOP(ok(z0)) → c14(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, c4, c5, c6, c7, c8, c9, c13, c14

61.31/21.61
61.31/21.61

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

Use narrowing to replace PROPER(f(z0, z1, z2)) → c9(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
61.31/21.61
61.31/21.61

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.61
active(c) → mark(a) 61.31/21.61
active(c) → mark(b) 61.31/21.61
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.61
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.61
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.61
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.61
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.61
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.61
proper(a) → ok(a) 61.31/21.61
proper(b) → ok(b) 61.31/21.61
proper(c) → ok(c) 61.31/21.61
top(mark(z0)) → top(proper(z0)) 61.31/21.61
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(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:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c13, c14, c9

61.31/21.61
61.31/21.61

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

Removed 9 trailing tuple parts
61.31/21.61
61.31/21.61

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.61
active(c) → mark(a) 61.31/21.61
active(c) → mark(b) 61.31/21.61
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.61
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.61
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.61
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.61
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.61
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.61
proper(a) → ok(a) 61.31/21.61
proper(b) → ok(b) 61.31/21.61
proper(c) → ok(c) 61.31/21.61
top(mark(z0)) → top(proper(z0)) 61.31/21.61
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c13, c14, c9, c9

61.31/21.61
61.31/21.61

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

Use narrowing to replace TOP(mark(z0)) → c13(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.61
TOP(mark(a)) → c13(TOP(ok(a)), PROPER(a)) 61.31/21.61
TOP(mark(b)) → c13(TOP(ok(b)), PROPER(b)) 61.31/21.61
TOP(mark(c)) → c13(TOP(ok(c)), PROPER(c))
61.31/21.61
61.31/21.61

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.61
active(c) → mark(a) 61.31/21.61
active(c) → mark(b) 61.31/21.61
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.61
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.61
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.61
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.61
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.61
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.61
proper(a) → ok(a) 61.31/21.61
proper(b) → ok(b) 61.31/21.61
proper(c) → ok(c) 61.31/21.61
top(mark(z0)) → top(proper(z0)) 61.31/21.61
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.61
TOP(mark(a)) → c13(TOP(ok(a)), PROPER(a)) 61.31/21.61
TOP(mark(b)) → c13(TOP(ok(b)), PROPER(b)) 61.31/21.61
TOP(mark(c)) → c13(TOP(ok(c)), PROPER(c))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.61
TOP(mark(a)) → c13(TOP(ok(a)), PROPER(a)) 61.31/21.61
TOP(mark(b)) → c13(TOP(ok(b)), PROPER(b)) 61.31/21.61
TOP(mark(c)) → c13(TOP(ok(c)), PROPER(c))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c14, c9, c9, c13

61.31/21.61
61.31/21.61

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

Removed 3 trailing tuple parts
61.31/21.61
61.31/21.61

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.61
active(c) → mark(a) 61.31/21.61
active(c) → mark(b) 61.31/21.61
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.61
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.61
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.61
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.61
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.61
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.61
proper(a) → ok(a) 61.31/21.61
proper(b) → ok(b) 61.31/21.61
proper(c) → ok(c) 61.31/21.61
top(mark(z0)) → top(proper(z0)) 61.31/21.61
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.61
TOP(mark(a)) → c13(TOP(ok(a))) 61.31/21.61
TOP(mark(b)) → c13(TOP(ok(b))) 61.31/21.61
TOP(mark(c)) → c13(TOP(ok(c)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.61
TOP(mark(a)) → c13(TOP(ok(a))) 61.31/21.61
TOP(mark(b)) → c13(TOP(ok(b))) 61.31/21.61
TOP(mark(c)) → c13(TOP(ok(c)))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c14, c9, c9, c13, c13

61.31/21.61
61.31/21.61

(11) 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)) → c13(TOP(ok(c)))
We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.61
proper(a) → ok(a) 61.31/21.61
proper(b) → ok(b) 61.31/21.61
proper(c) → ok(c) 61.31/21.61
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.61
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.61
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.61
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.61
active(c) → mark(a) 61.31/21.61
active(c) → mark(b) 61.31/21.61
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.61
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
And the Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.61
TOP(mark(a)) → c13(TOP(ok(a))) 61.31/21.61
TOP(mark(b)) → c13(TOP(ok(b))) 61.31/21.61
TOP(mark(c)) → c13(TOP(ok(c)))
The order we found is given by the following interpretation:
Polynomial interpretation : 61.31/21.61

POL(ACTIVE(x1)) = 0    61.31/21.61
POL(F(x1, x2, x3)) = 0    61.31/21.61
POL(PROPER(x1)) = 0    61.31/21.61
POL(TOP(x1)) = [2]x1    61.31/21.61
POL(a) = 0    61.31/21.61
POL(active(x1)) = 0    61.31/21.61
POL(b) = 0    61.31/21.61
POL(c) = [4]    61.31/21.61
POL(c1(x1)) = x1    61.31/21.61
POL(c13(x1)) = x1    61.31/21.61
POL(c13(x1, x2)) = x1 + x2    61.31/21.61
POL(c14(x1, x2)) = x1 + x2    61.31/21.61
POL(c4(x1, x2)) = x1 + x2    61.31/21.61
POL(c5(x1, x2)) = x1 + x2    61.31/21.61
POL(c6(x1)) = x1    61.31/21.61
POL(c7(x1)) = x1    61.31/21.61
POL(c8(x1)) = x1    61.31/21.61
POL(c9(x1, x2, x3)) = x1 + x2 + x3    61.31/21.61
POL(c9(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    61.31/21.61
POL(f(x1, x2, x3)) = 0    61.31/21.61
POL(mark(x1)) = x1    61.31/21.61
POL(ok(x1)) = 0    61.31/21.61
POL(proper(x1)) = 0   
61.31/21.61
61.31/21.61

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.61
active(c) → mark(a) 61.31/21.61
active(c) → mark(b) 61.31/21.61
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.61
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.61
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.61
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.61
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.61
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.61
proper(a) → ok(a) 61.31/21.61
proper(b) → ok(b) 61.31/21.61
proper(c) → ok(c) 61.31/21.61
top(mark(z0)) → top(proper(z0)) 61.31/21.61
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.61
TOP(mark(a)) → c13(TOP(ok(a))) 61.31/21.61
TOP(mark(b)) → c13(TOP(ok(b))) 61.31/21.61
TOP(mark(c)) → c13(TOP(ok(c)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.61
TOP(mark(a)) → c13(TOP(ok(a))) 61.31/21.61
TOP(mark(b)) → c13(TOP(ok(b)))
K tuples:

TOP(mark(c)) → c13(TOP(ok(c)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c14, c9, c9, c13, c13

61.31/21.61
61.31/21.61

(13) 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)) → c13(TOP(ok(b)))
We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.61
proper(a) → ok(a) 61.31/21.61
proper(b) → ok(b) 61.31/21.61
proper(c) → ok(c) 61.31/21.61
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.61
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.61
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.61
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.61
active(c) → mark(a) 61.31/21.61
active(c) → mark(b) 61.31/21.61
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.61
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
And the Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.61
TOP(mark(a)) → c13(TOP(ok(a))) 61.31/21.61
TOP(mark(b)) → c13(TOP(ok(b))) 61.31/21.61
TOP(mark(c)) → c13(TOP(ok(c)))
The order we found is given by the following interpretation:
Polynomial interpretation : 61.31/21.61

POL(ACTIVE(x1)) = 0    61.31/21.61
POL(F(x1, x2, x3)) = 0    61.31/21.61
POL(PROPER(x1)) = 0    61.31/21.61
POL(TOP(x1)) = [2]x1    61.31/21.61
POL(a) = [1]    61.31/21.61
POL(active(x1)) = x1    61.31/21.61
POL(b) = 0    61.31/21.61
POL(c) = [1]    61.31/21.61
POL(c1(x1)) = x1    61.31/21.61
POL(c13(x1)) = x1    61.31/21.61
POL(c13(x1, x2)) = x1 + x2    61.31/21.61
POL(c14(x1, x2)) = x1 + x2    61.31/21.61
POL(c4(x1, x2)) = x1 + x2    61.31/21.61
POL(c5(x1, x2)) = x1 + x2    61.31/21.61
POL(c6(x1)) = x1    61.31/21.61
POL(c7(x1)) = x1    61.31/21.61
POL(c8(x1)) = x1    61.31/21.61
POL(c9(x1, x2, x3)) = x1 + x2 + x3    61.31/21.61
POL(c9(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    61.31/21.61
POL(f(x1, x2, x3)) = [1]    61.31/21.61
POL(mark(x1)) = [1]    61.31/21.61
POL(ok(x1)) = x1    61.31/21.61
POL(proper(x1)) = 0   
61.31/21.61
61.31/21.61

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.61
active(c) → mark(a) 61.31/21.61
active(c) → mark(b) 61.31/21.61
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.61
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.61
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.61
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.61
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.61
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.61
proper(a) → ok(a) 61.31/21.61
proper(b) → ok(b) 61.31/21.61
proper(c) → ok(c) 61.31/21.61
top(mark(z0)) → top(proper(z0)) 61.31/21.61
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.61
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.61
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.61
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.61
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.61
TOP(mark(a)) → c13(TOP(ok(a))) 61.31/21.61
TOP(mark(b)) → c13(TOP(ok(b))) 61.31/21.61
TOP(mark(c)) → c13(TOP(ok(c)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.61
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.61
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.61
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.61
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.61
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.61
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.61
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.62
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.62
TOP(mark(a)) → c13(TOP(ok(a)))
K tuples:

TOP(mark(c)) → c13(TOP(ok(c))) 61.31/21.62
TOP(mark(b)) → c13(TOP(ok(b)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c14, c9, c9, c13, c13

61.31/21.62
61.31/21.62

(15) 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(a)) → c13(TOP(ok(a)))
We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.62
proper(a) → ok(a) 61.31/21.62
proper(b) → ok(b) 61.31/21.62
proper(c) → ok(c) 61.31/21.62
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.62
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.62
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.62
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.62
active(c) → mark(a) 61.31/21.62
active(c) → mark(b) 61.31/21.62
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.62
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
And the Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.62
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.62
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.62
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.62
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.62
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.62
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.62
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.62
TOP(mark(a)) → c13(TOP(ok(a))) 61.31/21.62
TOP(mark(b)) → c13(TOP(ok(b))) 61.31/21.62
TOP(mark(c)) → c13(TOP(ok(c)))
The order we found is given by the following interpretation:
Polynomial interpretation : 61.31/21.62

POL(ACTIVE(x1)) = 0    61.31/21.62
POL(F(x1, x2, x3)) = 0    61.31/21.62
POL(PROPER(x1)) = 0    61.31/21.62
POL(TOP(x1)) = [2]x1    61.31/21.62
POL(a) = 0    61.31/21.62
POL(active(x1)) = x1    61.31/21.62
POL(b) = 0    61.31/21.62
POL(c) = [1]    61.31/21.62
POL(c1(x1)) = x1    61.31/21.62
POL(c13(x1)) = x1    61.31/21.62
POL(c13(x1, x2)) = x1 + x2    61.31/21.62
POL(c14(x1, x2)) = x1 + x2    61.31/21.62
POL(c4(x1, x2)) = x1 + x2    61.31/21.62
POL(c5(x1, x2)) = x1 + x2    61.31/21.62
POL(c6(x1)) = x1    61.31/21.62
POL(c7(x1)) = x1    61.31/21.62
POL(c8(x1)) = x1    61.31/21.62
POL(c9(x1, x2, x3)) = x1 + x2 + x3    61.31/21.62
POL(c9(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    61.31/21.62
POL(f(x1, x2, x3)) = [1]    61.31/21.62
POL(mark(x1)) = [1]    61.31/21.62
POL(ok(x1)) = x1    61.31/21.62
POL(proper(x1)) = 0   
61.31/21.62
61.31/21.62

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.62
active(c) → mark(a) 61.31/21.62
active(c) → mark(b) 61.31/21.62
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.62
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.62
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.62
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.62
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.62
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.62
proper(a) → ok(a) 61.31/21.62
proper(b) → ok(b) 61.31/21.62
proper(c) → ok(c) 61.31/21.62
top(mark(z0)) → top(proper(z0)) 61.31/21.62
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.62
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.62
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.62
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.62
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.62
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.62
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.62
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.62
TOP(mark(a)) → c13(TOP(ok(a))) 61.31/21.62
TOP(mark(b)) → c13(TOP(ok(b))) 61.31/21.62
TOP(mark(c)) → c13(TOP(ok(c)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.62
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.62
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.62
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.62
TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) 61.31/21.62
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.62
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.62
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
K tuples:

TOP(mark(c)) → c13(TOP(ok(c))) 61.31/21.62
TOP(mark(b)) → c13(TOP(ok(b))) 61.31/21.62
TOP(mark(a)) → c13(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c4, c5, c6, c7, c8, c14, c9, c9, c13, c13

61.31/21.62
61.31/21.62

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

Use narrowing to replace TOP(ok(z0)) → c14(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(a, b, z0))) → c14(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(a)), ACTIVE(c)) 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(b)), ACTIVE(c)) 61.31/21.62
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(active(z0), z1, z2)), ACTIVE(f(z0, z1, z2))) 61.31/21.62
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
61.31/21.62
61.31/21.62

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.62
active(c) → mark(a) 61.31/21.62
active(c) → mark(b) 61.31/21.62
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.62
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.62
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.62
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.62
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.62
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.62
proper(a) → ok(a) 61.31/21.62
proper(b) → ok(b) 61.31/21.62
proper(c) → ok(c) 61.31/21.62
top(mark(z0)) → top(proper(z0)) 61.31/21.62
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.62
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.62
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.62
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.62
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.62
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.62
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.62
TOP(mark(a)) → c13(TOP(ok(a))) 61.31/21.62
TOP(mark(b)) → c13(TOP(ok(b))) 61.31/21.62
TOP(mark(c)) → c13(TOP(ok(c))) 61.31/21.62
TOP(ok(f(a, b, z0))) → c14(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(a)), ACTIVE(c)) 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(b)), ACTIVE(c)) 61.31/21.62
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(active(z0), z1, z2)), ACTIVE(f(z0, z1, z2))) 61.31/21.62
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.62
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.62
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.62
F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.62
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.62
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.62
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.62
TOP(ok(f(a, b, z0))) → c14(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(a)), ACTIVE(c)) 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(b)), ACTIVE(c)) 61.31/21.62
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(active(z0), z1, z2)), ACTIVE(f(z0, z1, z2))) 61.31/21.62
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
K tuples:

TOP(mark(c)) → c13(TOP(ok(c))) 61.31/21.62
TOP(mark(b)) → c13(TOP(ok(b))) 61.31/21.62
TOP(mark(a)) → c13(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9, c9, c13, c13, c14

61.31/21.62
61.31/21.62

(19) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 61.31/21.62
ACTIVE(f(z0, z1, z2)) → c4(F(active(z0), z1, z2), ACTIVE(z0)) 61.31/21.62
ACTIVE(f(z0, z1, z2)) → c5(F(z0, z1, active(z2)), ACTIVE(z2)) 61.31/21.62
PROPER(f(x0, x1, f(z0, z1, z2))) → c9(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 61.31/21.62
PROPER(f(x0, f(z0, z1, z2), x2)) → c9(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 61.31/21.62
PROPER(f(f(z0, z1, z2), x1, x2)) → c9(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(x0, x1, a)) → c9(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, b)) → c9(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, x1, c)) → c9(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 61.31/21.62
PROPER(f(x0, a, x2)) → c9(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, b, x2)) → c9(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(x0, c, x2)) → c9(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 61.31/21.62
PROPER(f(a, x1, x2)) → c9(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(b, x1, x2)) → c9(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
PROPER(f(c, x1, x2)) → c9(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 61.31/21.62
TOP(mark(f(z0, z1, z2))) → c13(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 61.31/21.62
TOP(ok(f(a, b, z0))) → c14(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 61.31/21.62
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(active(z0), z1, z2)), ACTIVE(f(z0, z1, z2))) 61.31/21.62
TOP(ok(f(z0, z1, z2))) → c14(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
61.31/21.62
61.31/21.62

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.62
active(c) → mark(a) 61.31/21.62
active(c) → mark(b) 61.31/21.62
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.62
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.62
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.62
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.62
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.62
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.62
proper(a) → ok(a) 61.31/21.62
proper(b) → ok(b) 61.31/21.62
proper(c) → ok(c) 61.31/21.62
top(mark(z0)) → top(proper(z0)) 61.31/21.62
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.62
TOP(mark(a)) → c13(TOP(ok(a))) 61.31/21.62
TOP(mark(b)) → c13(TOP(ok(b))) 61.31/21.62
TOP(mark(c)) → c13(TOP(ok(c))) 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(a)), ACTIVE(c)) 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(b)), ACTIVE(c))
S tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(a)), ACTIVE(c)) 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(b)), ACTIVE(c))
K tuples:

TOP(mark(c)) → c13(TOP(ok(c))) 61.31/21.62
TOP(mark(b)) → c13(TOP(ok(b))) 61.31/21.62
TOP(mark(a)) → c13(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP

Compound Symbols:

c6, c7, c8, c13, c14

61.31/21.62
61.31/21.62

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

Removed 4 trailing tuple parts
61.31/21.62
61.31/21.62

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.62
active(c) → mark(a) 61.31/21.62
active(c) → mark(b) 61.31/21.62
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.62
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.62
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.62
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.62
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.62
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.62
proper(a) → ok(a) 61.31/21.62
proper(b) → ok(b) 61.31/21.62
proper(c) → ok(c) 61.31/21.62
top(mark(z0)) → top(proper(z0)) 61.31/21.62
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.62
TOP(mark(c)) → c13(TOP(ok(c))) 61.31/21.62
TOP(mark(a)) → c13 61.31/21.62
TOP(mark(b)) → c13 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(a))) 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(b)))
S tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2)) 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(a))) 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(b)))
K tuples:

TOP(mark(c)) → c13(TOP(ok(c))) 61.31/21.62
TOP(mark(a)) → c13 61.31/21.62
TOP(mark(b)) → c13
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP

Compound Symbols:

c6, c7, c8, c13, c13, c14

61.31/21.62
61.31/21.62

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

Removed 5 trailing nodes:

TOP(ok(c)) → c14(TOP(mark(a))) 61.31/21.62
TOP(mark(b)) → c13 61.31/21.62
TOP(ok(c)) → c14(TOP(mark(b))) 61.31/21.62
TOP(mark(c)) → c13(TOP(ok(c))) 61.31/21.62
TOP(mark(a)) → c13
61.31/21.62
61.31/21.62

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.62
active(c) → mark(a) 61.31/21.62
active(c) → mark(b) 61.31/21.62
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.62
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.62
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.62
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.62
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.62
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.62
proper(a) → ok(a) 61.31/21.62
proper(b) → ok(b) 61.31/21.62
proper(c) → ok(c) 61.31/21.62
top(mark(z0)) → top(proper(z0)) 61.31/21.62
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
S tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c6, c7, c8

61.31/21.62
61.31/21.62

(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.

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

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 61.31/21.62

POL(F(x1, x2, x3)) = [2]x1 + x2    61.31/21.62
POL(c6(x1)) = x1    61.31/21.62
POL(c7(x1)) = x1    61.31/21.62
POL(c8(x1)) = x1    61.31/21.62
POL(mark(x1)) = [1] + x1    61.31/21.62
POL(ok(x1)) = x1   
61.31/21.62
61.31/21.62

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.62
active(c) → mark(a) 61.31/21.62
active(c) → mark(b) 61.31/21.62
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.62
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.62
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.62
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.62
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.62
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.62
proper(a) → ok(a) 61.31/21.62
proper(b) → ok(b) 61.31/21.62
proper(c) → ok(c) 61.31/21.62
top(mark(z0)) → top(proper(z0)) 61.31/21.62
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
S tuples:

F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
K tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c6, c7, c8

61.31/21.62
61.31/21.62

(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.

F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 61.31/21.62

POL(F(x1, x2, x3)) = x1 + x2 + x3    61.31/21.62
POL(c6(x1)) = x1    61.31/21.62
POL(c7(x1)) = x1    61.31/21.62
POL(c8(x1)) = x1    61.31/21.62
POL(mark(x1)) = [2] + x1    61.31/21.62
POL(ok(x1)) = x1   
61.31/21.62
61.31/21.62

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.62
active(c) → mark(a) 61.31/21.62
active(c) → mark(b) 61.31/21.62
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.62
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.62
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.62
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.62
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.62
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.62
proper(a) → ok(a) 61.31/21.62
proper(b) → ok(b) 61.31/21.62
proper(c) → ok(c) 61.31/21.62
top(mark(z0)) → top(proper(z0)) 61.31/21.62
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
S tuples:

F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
K tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c6, c7, c8

61.31/21.62
61.31/21.62

(29) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) 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)) → c8(F(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 61.31/21.62

POL(F(x1, x2, x3)) = x22    61.31/21.62
POL(c6(x1)) = x1    61.31/21.62
POL(c7(x1)) = x1    61.31/21.62
POL(c8(x1)) = x1    61.31/21.62
POL(mark(x1)) = [3]    61.31/21.62
POL(ok(x1)) = [1] + x1   
61.31/21.62
61.31/21.62

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 61.31/21.62
active(c) → mark(a) 61.31/21.62
active(c) → mark(b) 61.31/21.62
active(f(z0, z1, z2)) → f(active(z0), z1, z2) 61.31/21.62
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 61.31/21.62
f(mark(z0), z1, z2) → mark(f(z0, z1, z2)) 61.31/21.62
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 61.31/21.62
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 61.31/21.62
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 61.31/21.62
proper(a) → ok(a) 61.31/21.62
proper(b) → ok(b) 61.31/21.62
proper(c) → ok(c) 61.31/21.62
top(mark(z0)) → top(proper(z0)) 61.31/21.62
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
S tuples:none
K tuples:

F(mark(z0), z1, z2) → c6(F(z0, z1, z2)) 61.31/21.62
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2)) 61.31/21.62
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c6, c7, c8

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

The set S is empty
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(32) BOUNDS(O(1), O(1))

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61.31/21.67 EOF