YES(O(1), O(n^1)) 77.09/27.14 YES(O(1), O(n^1)) 77.09/27.16 77.09/27.16 77.09/27.16 77.09/27.16 77.09/27.16 77.09/27.16 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 77.09/27.16 77.09/27.16 77.09/27.16
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77.09/27.16
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
77.09/27.16 
↳2 CdtProblem
77.09/27.16 
↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
77.09/27.16 
↳4 CdtProblem
77.09/27.16 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
77.09/27.16 
↳6 CdtProblem
77.09/27.16 
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
77.09/27.16 
↳8 CdtProblem
77.09/27.16 
↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
77.09/27.16 
↳10 CdtProblem
77.09/27.16 
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
77.09/27.16 
↳12 CdtProblem
77.09/27.16 
↳13 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
77.09/27.16 
↳14 CdtProblem
77.09/27.16 
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
77.09/27.16 
↳16 CdtProblem
77.09/27.16 
↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
77.09/27.16 
↳18 CdtProblem
77.09/27.16 
↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
77.09/27.16 
↳20 CdtProblem
77.09/27.16 
↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
77.09/27.16 
↳22 CdtProblem
77.09/27.16 
↳23 CdtUnreachableProof (⇔)
77.09/27.16 
↳24 CdtProblem
77.09/27.16 
↳25 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
77.09/27.16 
↳26 CdtProblem
77.09/27.16 
↳27 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
77.09/27.16 
↳28 CdtProblem
77.09/27.16 
↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
77.09/27.16 
↳30 CdtProblem
77.09/27.16 
↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
77.09/27.16 
↳32 CdtProblem
77.09/27.16 
↳33 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳34 BOUNDS(O(1), O(1))
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77.09/27.16 77.09/27.16
77.09/27.16
77.09/27.16

(0) Obligation:

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

active(f(a, b, X)) → mark(f(X, X, X)) 77.09/27.16
active(c) → mark(a) 77.09/27.16
active(c) → mark(b) 77.09/27.16
active(f(X1, X2, X3)) → f(X1, X2, active(X3)) 77.09/27.16
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3)) 77.09/27.16
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3)) 77.09/27.16
proper(a) → ok(a) 77.09/27.16
proper(b) → ok(b) 77.09/27.16
proper(c) → ok(c) 77.09/27.16
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3)) 77.09/27.16
top(mark(X)) → top(proper(X)) 77.09/27.16
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
77.09/27.16
77.09/27.16

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

Converted CpxTRS to CDT
77.09/27.19
77.09/27.19

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.09/27.19
active(c) → mark(a) 77.09/27.19
active(c) → mark(b) 77.09/27.19
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.09/27.19
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.09/27.19
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.09/27.19
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.09/27.19
proper(a) → ok(a) 77.09/27.19
proper(b) → ok(b) 77.09/27.19
proper(c) → ok(c) 77.09/27.19
top(mark(z0)) → top(proper(z0)) 77.09/27.19
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.09/27.19
ACTIVE(f(z0, z1, z2)) → c4(F(z0, z1, active(z2)), ACTIVE(z2)) 77.09/27.19
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.09/27.19
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.09/27.19
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 77.09/27.19
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.09/27.19
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.09/27.19
ACTIVE(f(z0, z1, z2)) → c4(F(z0, z1, active(z2)), ACTIVE(z2)) 77.09/27.19
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.09/27.19
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.09/27.19
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 77.09/27.19
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.09/27.19
TOP(ok(z0)) → c12(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, c11, c12

77.09/27.19
77.09/27.19

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

Use narrowing to replace ACTIVE(f(z0, z1, z2)) → c4(F(z0, z1, active(z2)), ACTIVE(z2)) by

ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a)), ACTIVE(c)) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b)), ACTIVE(c)) 77.09/27.19
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
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77.09/27.19

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.09/27.19
active(c) → mark(a) 77.09/27.19
active(c) → mark(b) 77.09/27.19
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.09/27.19
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.09/27.19
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.09/27.19
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.09/27.19
proper(a) → ok(a) 77.09/27.19
proper(b) → ok(b) 77.09/27.19
proper(c) → ok(c) 77.09/27.19
top(mark(z0)) → top(proper(z0)) 77.09/27.19
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.09/27.19
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.09/27.19
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.09/27.19
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 77.09/27.19
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.09/27.19
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.09/27.19
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a)), ACTIVE(c)) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b)), ACTIVE(c)) 77.09/27.19
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.09/27.19
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.09/27.19
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.09/27.19
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 77.09/27.19
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.09/27.19
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.09/27.19
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a)), ACTIVE(c)) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b)), ACTIVE(c)) 77.09/27.19
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c5, c6, c7, c11, c12, c4

77.09/27.19
77.09/27.19

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

Removed 2 trailing tuple parts
77.09/27.19
77.09/27.19

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.09/27.19
active(c) → mark(a) 77.09/27.19
active(c) → mark(b) 77.09/27.19
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.09/27.19
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.09/27.19
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.09/27.19
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.09/27.19
proper(a) → ok(a) 77.09/27.19
proper(b) → ok(b) 77.09/27.19
proper(c) → ok(c) 77.09/27.19
top(mark(z0)) → top(proper(z0)) 77.09/27.19
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.09/27.19
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.09/27.19
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.09/27.19
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 77.09/27.19
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.09/27.19
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.09/27.19
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.09/27.19
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.09/27.19
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.09/27.19
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.09/27.19
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 77.09/27.19
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.09/27.19
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.09/27.19
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.09/27.19
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b)))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c5, c6, c7, c11, c12, c4, c4

77.09/27.19
77.09/27.19

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

Use narrowing to replace PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a)) 77.40/27.22
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 77.40/27.22
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2)) 77.40/27.22
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 77.40/27.22
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
77.40/27.22
77.40/27.22

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.22
active(c) → mark(a) 77.40/27.22
active(c) → mark(b) 77.40/27.22
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.22
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.22
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.22
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.22
proper(a) → ok(a) 77.40/27.22
proper(b) → ok(b) 77.40/27.22
proper(c) → ok(c) 77.40/27.22
top(mark(z0)) → top(proper(z0)) 77.40/27.22
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.22
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.22
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.22
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.40/27.22
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.22
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.22
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.22
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a)) 77.40/27.22
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 77.40/27.22
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2)) 77.40/27.22
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 77.40/27.22
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(c, x1, x2)) → c7(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)) 77.40/27.22
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.22
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.22
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.40/27.22
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.22
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.22
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.22
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a)) 77.40/27.22
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 77.40/27.22
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2)) 77.40/27.22
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 77.40/27.22
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(c, x1, x2)) → c7(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, c5, c6, c11, c12, c4, c4, c7

77.40/27.22
77.40/27.22

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

Removed 9 trailing tuple parts
77.40/27.22
77.40/27.22

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.22
active(c) → mark(a) 77.40/27.22
active(c) → mark(b) 77.40/27.22
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.22
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.22
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.22
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.22
proper(a) → ok(a) 77.40/27.22
proper(b) → ok(b) 77.40/27.22
proper(c) → ok(c) 77.40/27.22
top(mark(z0)) → top(proper(z0)) 77.40/27.22
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.22
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.22
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.22
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.40/27.22
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.22
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.22
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.22
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.22
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.22
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.22
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.40/27.22
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.22
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.22
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.22
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(c, x1, x2)) → c7(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, c5, c6, c11, c12, c4, c4, c7, c7

77.40/27.22
77.40/27.22

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

Use narrowing to replace TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.22
TOP(mark(a)) → c11(TOP(ok(a)), PROPER(a)) 77.40/27.22
TOP(mark(b)) → c11(TOP(ok(b)), PROPER(b)) 77.40/27.22
TOP(mark(c)) → c11(TOP(ok(c)), PROPER(c))
77.40/27.22
77.40/27.22

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.22
active(c) → mark(a) 77.40/27.22
active(c) → mark(b) 77.40/27.22
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.22
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.22
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.22
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.22
proper(a) → ok(a) 77.40/27.22
proper(b) → ok(b) 77.40/27.22
proper(c) → ok(c) 77.40/27.22
top(mark(z0)) → top(proper(z0)) 77.40/27.22
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.22
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.22
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.22
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.22
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.22
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.22
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.22
TOP(mark(a)) → c11(TOP(ok(a)), PROPER(a)) 77.40/27.22
TOP(mark(b)) → c11(TOP(ok(b)), PROPER(b)) 77.40/27.22
TOP(mark(c)) → c11(TOP(ok(c)), PROPER(c))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.22
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.22
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.22
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.22
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.22
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.22
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a)), PROPER(a)) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b)), PROPER(b)) 77.40/27.23
TOP(mark(c)) → c11(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, c5, c6, c12, c4, c4, c7, c7, c11

77.40/27.23
77.40/27.23

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

Removed 3 trailing tuple parts
77.40/27.23
77.40/27.23

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c12, c4, c4, c7, c7, c11, c11

77.40/27.23
77.40/27.23

(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(b)) → c11(TOP(ok(b)))
We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
And the Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
The order we found is given by the following interpretation:
Polynomial interpretation : 77.40/27.23

POL(ACTIVE(x1)) = 0    77.40/27.23
POL(F(x1, x2, x3)) = 0    77.40/27.23
POL(PROPER(x1)) = 0    77.40/27.23
POL(TOP(x1)) = x1    77.40/27.23
POL(a) = [2]    77.40/27.23
POL(active(x1)) = x1    77.40/27.23
POL(b) = 0    77.40/27.23
POL(c) = [2]    77.40/27.23
POL(c1(x1)) = x1    77.40/27.23
POL(c11(x1)) = x1    77.40/27.23
POL(c11(x1, x2)) = x1 + x2    77.40/27.23
POL(c12(x1, x2)) = x1 + x2    77.40/27.23
POL(c4(x1)) = x1    77.40/27.23
POL(c4(x1, x2)) = x1 + x2    77.40/27.23
POL(c5(x1)) = x1    77.40/27.23
POL(c6(x1)) = x1    77.40/27.23
POL(c7(x1, x2, x3)) = x1 + x2 + x3    77.40/27.23
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    77.40/27.23
POL(f(x1, x2, x3)) = [2]    77.40/27.23
POL(mark(x1)) = [2]    77.40/27.23
POL(ok(x1)) = x1    77.40/27.23
POL(proper(x1)) = [2]x1   
77.40/27.23
77.40/27.23

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
K tuples:

TOP(mark(b)) → c11(TOP(ok(b)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c12, c4, c4, c7, c7, c11, c11

77.40/27.23
77.40/27.23

(17) 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)) → c11(TOP(ok(c)))
We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
And the Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
The order we found is given by the following interpretation:
Polynomial interpretation : 77.40/27.23

POL(ACTIVE(x1)) = 0    77.40/27.23
POL(F(x1, x2, x3)) = 0    77.40/27.23
POL(PROPER(x1)) = 0    77.40/27.23
POL(TOP(x1)) = [2]x1    77.40/27.23
POL(a) = 0    77.40/27.23
POL(active(x1)) = 0    77.40/27.23
POL(b) = 0    77.40/27.23
POL(c) = [2]    77.40/27.23
POL(c1(x1)) = x1    77.40/27.23
POL(c11(x1)) = x1    77.40/27.23
POL(c11(x1, x2)) = x1 + x2    77.40/27.23
POL(c12(x1, x2)) = x1 + x2    77.40/27.23
POL(c4(x1)) = x1    77.40/27.23
POL(c4(x1, x2)) = x1 + x2    77.40/27.23
POL(c5(x1)) = x1    77.40/27.23
POL(c6(x1)) = x1    77.40/27.23
POL(c7(x1, x2, x3)) = x1 + x2 + x3    77.40/27.23
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    77.40/27.23
POL(f(x1, x2, x3)) = 0    77.40/27.23
POL(mark(x1)) = x1    77.40/27.23
POL(ok(x1)) = 0    77.40/27.23
POL(proper(x1)) = 0   
77.40/27.23
77.40/27.23

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a)))
K tuples:

TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c12, c4, c4, c7, c7, c11, c11

77.40/27.23
77.40/27.23

(19) 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)) → c11(TOP(ok(a)))
We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
And the Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
The order we found is given by the following interpretation:
Polynomial interpretation : 77.40/27.23

POL(ACTIVE(x1)) = 0    77.40/27.23
POL(F(x1, x2, x3)) = 0    77.40/27.23
POL(PROPER(x1)) = 0    77.40/27.23
POL(TOP(x1)) = [4]x1    77.40/27.23
POL(a) = 0    77.40/27.23
POL(active(x1)) = x1    77.40/27.23
POL(b) = 0    77.40/27.23
POL(c) = [2]    77.40/27.23
POL(c1(x1)) = x1    77.40/27.23
POL(c11(x1)) = x1    77.40/27.23
POL(c11(x1, x2)) = x1 + x2    77.40/27.23
POL(c12(x1, x2)) = x1 + x2    77.40/27.23
POL(c4(x1)) = x1    77.40/27.23
POL(c4(x1, x2)) = x1 + x2    77.40/27.23
POL(c5(x1)) = x1    77.40/27.23
POL(c6(x1)) = x1    77.40/27.23
POL(c7(x1, x2, x3)) = x1 + x2 + x3    77.40/27.23
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    77.40/27.23
POL(f(x1, x2, x3)) = [2]    77.40/27.23
POL(mark(x1)) = [2]    77.40/27.23
POL(ok(x1)) = x1    77.40/27.23
POL(proper(x1)) = 0   
77.40/27.23
77.40/27.23

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
K tuples:

TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c12, c4, c4, c7, c7, c11, c11

77.40/27.23
77.40/27.23

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

Use narrowing to replace TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(a, b, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a)), ACTIVE(c)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)), ACTIVE(c)) 77.40/27.23
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
77.40/27.23
77.40/27.23

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(ok(f(a, b, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a)), ACTIVE(c)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)), ACTIVE(c)) 77.40/27.23
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
S tuples:

ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(ok(f(a, b, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a)), ACTIVE(c)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)), ACTIVE(c)) 77.40/27.23
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
K tuples:

TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c1, c5, c6, c4, c4, c7, c7, c11, c11, c12

77.40/27.23
77.40/27.23

(23) 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)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(ok(f(a, b, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
77.40/27.23
77.40/27.23

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a)), ACTIVE(c)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)), ACTIVE(c))
S tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a)), ACTIVE(c)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)), ACTIVE(c))
K tuples:

TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP

Compound Symbols:

c5, c6, c11, c12

77.40/27.23
77.40/27.23

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

Removed 4 trailing tuple parts
77.40/27.23
77.40/27.23

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(mark(a)) → c11 77.40/27.23
TOP(mark(b)) → c11 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)))
S tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)))
K tuples:

TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(mark(a)) → c11 77.40/27.23
TOP(mark(b)) → c11
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP

Compound Symbols:

c5, c6, c11, c11, c12

77.40/27.23
77.40/27.23

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

Removed 5 trailing nodes:

TOP(ok(c)) → c12(TOP(mark(b))) 77.40/27.23
TOP(mark(b)) → c11 77.40/27.23
TOP(mark(a)) → c11 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a)))
77.40/27.23
77.40/27.23

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
S tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c5, c6

77.40/27.23
77.40/27.23

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

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 77.40/27.23

POL(F(x1, x2, x3)) = x1 + x2 + [2]x3    77.40/27.23
POL(c5(x1)) = x1    77.40/27.23
POL(c6(x1)) = x1    77.40/27.23
POL(mark(x1)) = [1] + x1    77.40/27.23
POL(ok(x1)) = x1   
77.40/27.23
77.40/27.23

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
S tuples:

F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
K tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c5, c6

77.40/27.23
77.40/27.23

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

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 77.40/27.23

POL(F(x1, x2, x3)) = x1    77.40/27.23
POL(c5(x1)) = x1    77.40/27.23
POL(c6(x1)) = x1    77.40/27.23
POL(mark(x1)) = x1    77.40/27.23
POL(ok(x1)) = [1] + x1   
77.40/27.23
77.40/27.23

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
Tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
S tuples:none
K tuples:

F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c5, c6

77.40/27.23
77.40/27.23

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

The set S is empty
77.40/27.23
77.40/27.23

(34) BOUNDS(O(1), O(1))

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77.40/27.23
77.65/27.30 EOF