YES(O(1), O(n^1)) 61.04/21.85 YES(O(1), O(n^1)) 61.04/21.88 61.04/21.88 61.04/21.88 61.04/21.88 61.04/21.88 61.04/21.88 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 61.04/21.88 61.04/21.88 61.04/21.88
61.04/21.88
61.04/21.88
61.04/21.88
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
61.04/21.88 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳2 CdtProblem
61.04/21.88 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳4 CdtProblem
61.04/21.88 
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳6 CdtProblem
61.04/21.88 
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳8 CdtProblem
61.04/21.88 
↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳10 CdtProblem
61.04/21.88 
↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳12 CdtProblem
61.04/21.88 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
61.04/21.88 
↳14 CdtProblem
61.04/21.88 
↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳16 CdtProblem
61.04/21.88 
↳17 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳18 CdtProblem
61.04/21.88 
↳19 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳20 CdtProblem
61.04/21.88 
↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳22 CdtProblem
61.04/21.88 
↳23 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳24 CdtProblem
61.04/21.88 
↳25 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳26 CdtProblem
61.04/21.88 
↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
61.04/21.88 
↳28 CdtProblem
61.04/21.88 
↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
61.04/21.88 
↳30 CdtProblem
61.04/21.88 
↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳32 CdtProblem
61.04/21.88 
↳33 CdtUnreachableProof (⇔)
61.04/21.88 
↳34 CdtProblem
61.04/21.88 
↳35 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳36 CdtProblem
61.04/21.88 
↳37 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳38 CdtProblem
61.04/21.88 
↳39 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
61.04/21.88 
↳40 CdtProblem
61.04/21.88 
↳41 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
61.04/21.88 
↳42 CdtProblem
61.04/21.88 
↳43 SIsEmptyProof (BOTH BOUNDS(ID, ID))
61.04/21.88 
↳44 BOUNDS(O(1), O(1))
61.04/21.88
61.04/21.88 61.04/21.88
61.04/21.88
61.04/21.88

(0) Obligation:

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

active(f(X, X)) → mark(f(a, b)) 61.04/21.88
active(b) → mark(a) 61.04/21.88
active(f(X1, X2)) → f(active(X1), X2) 61.04/21.88
f(mark(X1), X2) → mark(f(X1, X2)) 61.04/21.88
proper(f(X1, X2)) → f(proper(X1), proper(X2)) 61.04/21.88
proper(a) → ok(a) 61.04/21.88
proper(b) → ok(b) 61.04/21.88
f(ok(X1), ok(X2)) → ok(f(X1, X2)) 61.04/21.88
top(mark(X)) → top(proper(X)) 61.04/21.88
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
61.35/21.91
61.35/21.91

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

Converted CpxTRS to CDT
61.35/21.91
61.35/21.91

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.91
active(b) → mark(a) 61.35/21.91
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.91
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.91
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.91
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.91
proper(a) → ok(a) 61.35/21.91
proper(b) → ok(b) 61.35/21.91
top(mark(z0)) → top(proper(z0)) 61.35/21.91
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, z0)) → c(F(a, b)) 61.35/21.91
ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0)) 61.35/21.91
F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.91
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.91
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.91
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.91
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0, z0)) → c(F(a, b)) 61.35/21.91
ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0)) 61.35/21.91
F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.91
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.91
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.91
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.91
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c, c2, c3, c4, c5, c8, c9

61.35/21.91
61.35/21.91

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

Removed 1 trailing tuple parts
61.35/21.91
61.35/21.91

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.91
active(b) → mark(a) 61.35/21.91
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.91
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.91
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.91
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.91
proper(a) → ok(a) 61.35/21.91
proper(b) → ok(b) 61.35/21.91
top(mark(z0)) → top(proper(z0)) 61.35/21.91
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0)) 61.35/21.91
F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.91
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.91
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.91
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.91
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.91
ACTIVE(f(z0, z0)) → c
S tuples:

ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0)) 61.35/21.91
F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.91
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.91
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.91
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.91
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.91
ACTIVE(f(z0, z0)) → c
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c2, c3, c4, c5, c8, c9, c

61.35/21.91
61.35/21.91

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

Removed 1 trailing nodes:

ACTIVE(f(z0, z0)) → c
61.35/21.93
61.35/21.93

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.93
active(b) → mark(a) 61.35/21.93
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.93
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.93
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.93
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.93
proper(a) → ok(a) 61.35/21.93
proper(b) → ok(b) 61.35/21.93
top(mark(z0)) → top(proper(z0)) 61.35/21.93
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0)) 61.35/21.93
F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.93
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.93
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.93
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.93
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.93
ACTIVE(f(z0, z0)) → c
S tuples:

ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0)) 61.35/21.93
F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.93
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.93
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.93
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.93
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.93
ACTIVE(f(z0, z0)) → c
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c2, c3, c4, c5, c8, c9, c

61.35/21.93
61.35/21.93

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

Use narrowing to replace ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0)) by

ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.93
ACTIVE(f(b, x1)) → c2(F(mark(a), x1), ACTIVE(b)) 61.35/21.93
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
61.35/21.93
61.35/21.93

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.93
active(b) → mark(a) 61.35/21.93
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.93
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.93
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.93
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.93
proper(a) → ok(a) 61.35/21.93
proper(b) → ok(b) 61.35/21.93
top(mark(z0)) → top(proper(z0)) 61.35/21.93
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.93
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.93
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.93
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.93
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.93
ACTIVE(f(z0, z0)) → c 61.35/21.93
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.93
ACTIVE(f(b, x1)) → c2(F(mark(a), x1), ACTIVE(b)) 61.35/21.93
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.93
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.93
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.93
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.93
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.93
ACTIVE(f(z0, z0)) → c 61.35/21.93
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.93
ACTIVE(f(b, x1)) → c2(F(mark(a), x1), ACTIVE(b)) 61.35/21.93
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c8, c9, c, c2

61.35/21.93
61.35/21.93

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

Removed 1 trailing tuple parts
61.35/21.93
61.35/21.93

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.93
active(b) → mark(a) 61.35/21.93
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.93
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.93
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.93
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.93
proper(a) → ok(a) 61.35/21.93
proper(b) → ok(b) 61.35/21.93
top(mark(z0)) → top(proper(z0)) 61.35/21.93
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.93
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.93
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.93
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.93
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.93
ACTIVE(f(z0, z0)) → c 61.35/21.93
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.93
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.93
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.93
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.93
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.93
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.93
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.93
ACTIVE(f(z0, z0)) → c 61.35/21.93
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.93
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.93
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c8, c9, c, c2, c2

61.35/21.93
61.35/21.93

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

Removed 1 trailing nodes:

ACTIVE(f(z0, z0)) → c
61.35/21.93
61.35/21.93

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.93
active(b) → mark(a) 61.35/21.93
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.93
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.93
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.93
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.93
proper(a) → ok(a) 61.35/21.93
proper(b) → ok(b) 61.35/21.93
top(mark(z0)) → top(proper(z0)) 61.35/21.93
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.93
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.93
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.93
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.93
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.93
ACTIVE(f(z0, z0)) → c 61.35/21.93
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.93
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.93
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.93
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.93
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.93
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.93
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.93
ACTIVE(f(z0, z0)) → c 61.35/21.93
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.93
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.93
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c8, c9, c, c2, c2

61.35/21.93
61.35/21.93

(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.

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
We considered the (Usable) Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.93
active(b) → mark(a) 61.35/21.93
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.93
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.93
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.93
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.93
proper(a) → ok(a) 61.35/21.93
proper(b) → ok(b)
And the Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.93
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.93
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.93
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.93
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.93
ACTIVE(f(z0, z0)) → c 61.35/21.93
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.93
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.93
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
The order we found is given by the following interpretation:
Polynomial interpretation : 61.35/21.93

POL(ACTIVE(x1)) = x1    61.35/21.93
POL(F(x1, x2)) = 0    61.35/21.93
POL(PROPER(x1)) = 0    61.35/21.93
POL(TOP(x1)) = x1    61.35/21.93
POL(a) = 0    61.35/21.93
POL(active(x1)) = 0    61.35/21.93
POL(b) = [4]    61.35/21.93
POL(c) = 0    61.35/21.93
POL(c2(x1)) = x1    61.35/21.93
POL(c2(x1, x2)) = x1 + x2    61.35/21.93
POL(c3(x1)) = x1    61.35/21.93
POL(c4(x1)) = x1    61.35/21.93
POL(c5(x1, x2, x3)) = x1 + x2 + x3    61.35/21.93
POL(c8(x1, x2)) = x1 + x2    61.35/21.93
POL(c9(x1, x2)) = x1 + x2    61.35/21.93
POL(f(x1, x2)) = [4]x1    61.35/21.93
POL(mark(x1)) = x1    61.35/21.93
POL(ok(x1)) = x1    61.35/21.93
POL(proper(x1)) = x1   
61.35/21.93
61.35/21.93

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.93
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.95
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 61.35/21.95
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c8, c9, c, c2, c2

61.35/21.95
61.35/21.95

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

Use narrowing to replace PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0), PROPER(a)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0), PROPER(b)) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(a), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
61.35/21.95
61.35/21.95

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0), PROPER(a)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0), PROPER(b)) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(a), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0), PROPER(a)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0), PROPER(b)) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(a), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c8, c9, c, c2, c2, c5

61.35/21.95
61.35/21.95

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

Removed 4 trailing tuple parts
61.35/21.95
61.35/21.95

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c8, c9, c, c2, c2, c5, c5

61.35/21.95
61.35/21.95

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

Removed 1 trailing nodes:

ACTIVE(f(z0, z0)) → c
61.35/21.95
61.35/21.95

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c8, c9, c, c2, c2, c5, c5

61.35/21.95
61.35/21.95

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

Use narrowing to replace TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a)) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b)), PROPER(b))
61.35/21.95
61.35/21.95

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a)) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b)), PROPER(b))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a)) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b)), PROPER(b))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c9, c, c2, c2, c5, c5, c8

61.35/21.95
61.35/21.95

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

Removed 2 trailing tuple parts
61.35/21.95
61.35/21.95

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a))) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a))) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b)))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c9, c, c2, c2, c5, c5, c8, c8

61.35/21.95
61.35/21.95

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

Removed 1 trailing nodes:

ACTIVE(f(z0, z0)) → c
61.35/21.95
61.35/21.95

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a))) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a))) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b)))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c9, c, c2, c2, c5, c5, c8, c8

61.35/21.95
61.35/21.95

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

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

TOP(mark(b)) → c8(TOP(ok(b)))
We considered the (Usable) Rules:

proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1)
And the Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a))) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b)))
The order we found is given by the following interpretation:
Polynomial interpretation : 61.35/21.95

POL(ACTIVE(x1)) = 0    61.35/21.95
POL(F(x1, x2)) = 0    61.35/21.95
POL(PROPER(x1)) = 0    61.35/21.95
POL(TOP(x1)) = x1    61.35/21.95
POL(a) = 0    61.35/21.95
POL(active(x1)) = 0    61.35/21.95
POL(b) = [2]    61.35/21.95
POL(c) = 0    61.35/21.95
POL(c2(x1)) = x1    61.35/21.95
POL(c2(x1, x2)) = x1 + x2    61.35/21.95
POL(c3(x1)) = x1    61.35/21.95
POL(c4(x1)) = x1    61.35/21.95
POL(c5(x1, x2)) = x1 + x2    61.35/21.95
POL(c5(x1, x2, x3)) = x1 + x2 + x3    61.35/21.95
POL(c8(x1)) = x1    61.35/21.95
POL(c8(x1, x2)) = x1 + x2    61.35/21.95
POL(c9(x1, x2)) = x1 + x2    61.35/21.95
POL(f(x1, x2)) = 0    61.35/21.95
POL(mark(x1)) = x1    61.35/21.95
POL(ok(x1)) = 0    61.35/21.95
POL(proper(x1)) = 0   
61.35/21.95
61.35/21.95

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a))) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a)))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c9, c, c2, c2, c5, c5, c8, c8

61.35/21.95
61.35/21.95

(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.

TOP(mark(a)) → c8(TOP(ok(a)))
We considered the (Usable) Rules:

proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1)
And the Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a))) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b)))
The order we found is given by the following interpretation:
Polynomial interpretation : 61.35/21.95

POL(ACTIVE(x1)) = 0    61.35/21.95
POL(F(x1, x2)) = 0    61.35/21.95
POL(PROPER(x1)) = 0    61.35/21.95
POL(TOP(x1)) = [2]x1    61.35/21.95
POL(a) = 0    61.35/21.95
POL(active(x1)) = x1    61.35/21.95
POL(b) = [4]    61.35/21.95
POL(c) = 0    61.35/21.95
POL(c2(x1)) = x1    61.35/21.95
POL(c2(x1, x2)) = x1 + x2    61.35/21.95
POL(c3(x1)) = x1    61.35/21.95
POL(c4(x1)) = x1    61.35/21.95
POL(c5(x1, x2)) = x1 + x2    61.35/21.95
POL(c5(x1, x2, x3)) = x1 + x2 + x3    61.35/21.95
POL(c8(x1)) = x1    61.35/21.95
POL(c8(x1, x2)) = x1 + x2    61.35/21.95
POL(c9(x1, x2)) = x1 + x2    61.35/21.95
POL(f(x1, x2)) = [4]    61.35/21.95
POL(mark(x1)) = [4]    61.35/21.95
POL(ok(x1)) = x1    61.35/21.95
POL(proper(x1)) = 0   
61.35/21.95
61.35/21.95

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a))) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c9, c, c2, c2, c5, c5, c8, c8

61.35/21.95
61.35/21.95

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

Use narrowing to replace TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0))) 61.35/21.95
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b)) 61.35/21.95
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
61.35/21.95
61.35/21.95

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a))) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b))) 61.35/21.95
TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0))) 61.35/21.95
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b)) 61.35/21.95
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0))) 61.35/21.95
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b)) 61.35/21.95
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
K tuples:

ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, ACTIVE, PROPER, TOP

Compound Symbols:

c3, c4, c, c2, c2, c5, c5, c8, c8, c9

61.35/21.95
61.35/21.95

(33) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(f(z0, z0)) → c 61.35/21.95
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0))) 61.35/21.95
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 61.35/21.95
ACTIVE(f(b, x1)) → c2(F(mark(a), x1)) 61.35/21.95
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 61.35/21.95
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 61.35/21.95
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0)) 61.35/21.95
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0)) 61.35/21.95
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1)) 61.35/21.95
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1)) 61.35/21.95
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 61.35/21.95
TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0))) 61.35/21.95
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
61.35/21.95
61.35/21.95

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a))) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b))) 61.35/21.95
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))
K tuples:

TOP(mark(b)) → c8(TOP(ok(b))) 61.35/21.95
TOP(mark(a)) → c8(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP

Compound Symbols:

c3, c4, c8, c9

61.35/21.95
61.35/21.95

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

Removed 2 trailing tuple parts
61.35/21.95
61.35/21.95

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b))) 61.35/21.95
TOP(mark(a)) → c8 61.35/21.95
TOP(ok(b)) → c9(TOP(mark(a)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1)) 61.35/21.95
TOP(ok(b)) → c9(TOP(mark(a)))
K tuples:

TOP(mark(b)) → c8(TOP(ok(b))) 61.35/21.95
TOP(mark(a)) → c8
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP

Compound Symbols:

c3, c4, c8, c8, c9

61.35/21.95
61.35/21.95

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

Removed 3 trailing nodes:

TOP(mark(a)) → c8 61.35/21.95
TOP(ok(b)) → c9(TOP(mark(a))) 61.35/21.95
TOP(mark(b)) → c8(TOP(ok(b)))
61.35/21.95
61.35/21.95

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

61.35/21.95
61.35/21.95

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

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

F(mark(z0), z1) → c3(F(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 61.35/21.95

POL(F(x1, x2)) = [4]x1    61.35/21.95
POL(c3(x1)) = x1    61.35/21.95
POL(c4(x1)) = x1    61.35/21.95
POL(mark(x1)) = [4] + x1    61.35/21.95
POL(ok(x1)) = x1   
61.35/21.95
61.35/21.95

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1))
S tuples:

F(ok(z0), ok(z1)) → c4(F(z0, z1))
K tuples:

F(mark(z0), z1) → c3(F(z0, z1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

61.35/21.95
61.35/21.95

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

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 61.35/21.95

POL(F(x1, x2)) = x1    61.35/21.95
POL(c3(x1)) = x1    61.35/21.95
POL(c4(x1)) = x1    61.35/21.95
POL(mark(x1)) = x1    61.35/21.95
POL(ok(x1)) = [1] + x1   
61.35/21.95
61.35/21.95

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b)) 61.35/21.95
active(b) → mark(a) 61.35/21.95
active(f(z0, z1)) → f(active(z0), z1) 61.35/21.95
f(mark(z0), z1) → mark(f(z0, z1)) 61.35/21.95
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 61.35/21.95
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 61.35/21.95
proper(a) → ok(a) 61.35/21.95
proper(b) → ok(b) 61.35/21.95
top(mark(z0)) → top(proper(z0)) 61.35/21.95
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1))
S tuples:none
K tuples:

F(mark(z0), z1) → c3(F(z0, z1)) 61.35/21.95
F(ok(z0), ok(z1)) → c4(F(z0, z1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

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

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

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61.61/22.07 EOF