YES(O(1), O(n^1)) 96.12/33.42 YES(O(1), O(n^1)) 96.12/33.45 96.12/33.45 96.12/33.45 96.12/33.45 96.12/33.45 96.12/33.45 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 96.12/33.45 96.12/33.45 96.12/33.45
96.12/33.45
96.12/33.45
96.12/33.45
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
96.12/33.45 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳2 CdtProblem
96.12/33.45 
↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳4 CdtProblem
96.12/33.45 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳6 CdtProblem
96.12/33.45 
↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳8 CdtProblem
96.12/33.45 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
96.12/33.45 
↳10 CdtProblem
96.12/33.45 
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳12 CdtProblem
96.12/33.45 
↳13 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳14 CdtProblem
96.12/33.45 
↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳16 CdtProblem
96.12/33.45 
↳17 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳18 CdtProblem
96.12/33.45 
↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳20 CdtProblem
96.12/33.45 
↳21 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳22 CdtProblem
96.12/33.45 
↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
96.12/33.45 
↳24 CdtProblem
96.12/33.45 
↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
96.12/33.45 
↳26 CdtProblem
96.12/33.45 
↳27 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳28 CdtProblem
96.12/33.45 
↳29 CdtUnreachableProof (⇔)
96.12/33.45 
↳30 CdtProblem
96.12/33.45 
↳31 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳32 CdtProblem
96.12/33.45 
↳33 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳34 CdtProblem
96.12/33.45 
↳35 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
96.12/33.45 
↳36 CdtProblem
96.12/33.45 
↳37 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
96.12/33.45 
↳38 CdtProblem
96.12/33.45 
↳39 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
96.12/33.45 
↳40 CdtProblem
96.12/33.45 
↳41 SIsEmptyProof (BOTH BOUNDS(ID, ID))
96.12/33.45 
↳42 BOUNDS(O(1), O(1))
96.12/33.45
96.12/33.45 96.12/33.45
96.12/33.45
96.12/33.45

(0) Obligation:

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

active(f(X, g(X), Y)) → mark(f(Y, Y, Y)) 96.12/33.45
active(g(b)) → mark(c) 96.12/33.45
active(b) → mark(c) 96.12/33.45
active(g(X)) → g(active(X)) 96.12/33.45
g(mark(X)) → mark(g(X)) 96.12/33.45
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3)) 96.12/33.45
proper(g(X)) → g(proper(X)) 96.12/33.45
proper(b) → ok(b) 96.12/33.45
proper(c) → ok(c) 96.12/33.45
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3)) 96.47/33.50
g(ok(X)) → ok(g(X)) 96.47/33.50
top(mark(X)) → top(proper(X)) 96.47/33.50
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
96.47/33.50
96.47/33.50

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

Converted CpxTRS to CDT
96.47/33.50
96.47/33.50

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.50
active(g(b)) → mark(c) 96.47/33.50
active(b) → mark(c) 96.47/33.50
active(g(z0)) → g(active(z0)) 96.47/33.50
g(mark(z0)) → mark(g(z0)) 96.47/33.50
g(ok(z0)) → ok(g(z0)) 96.47/33.50
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.50
proper(g(z0)) → g(proper(z0)) 96.47/33.50
proper(b) → ok(b) 96.47/33.50
proper(c) → ok(c) 96.47/33.50
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.50
top(mark(z0)) → top(proper(z0)) 96.47/33.50
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.50
ACTIVE(g(z0)) → c4(G(active(z0)), ACTIVE(z0)) 96.47/33.50
G(mark(z0)) → c5(G(z0)) 96.47/33.50
G(ok(z0)) → c6(G(z0)) 96.47/33.50
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.47/33.50
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.50
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.50
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.50
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.50
ACTIVE(g(z0)) → c4(G(active(z0)), ACTIVE(z0)) 96.47/33.50
G(mark(z0)) → c5(G(z0)) 96.47/33.50
G(ok(z0)) → c6(G(z0)) 96.47/33.50
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.47/33.50
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.50
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.50
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.50
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c4, c5, c6, c7, c8, c11, c12, c13

96.47/33.50
96.47/33.50

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

Use narrowing to replace ACTIVE(g(z0)) → c4(G(active(z0)), ACTIVE(z0)) by

ACTIVE(g(f(z0, g(z0), z1))) → c4(G(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1))) 96.47/33.50
ACTIVE(g(g(b))) → c4(G(mark(c)), ACTIVE(g(b))) 96.47/33.50
ACTIVE(g(b)) → c4(G(mark(c)), ACTIVE(b)) 96.47/33.50
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
96.47/33.50
96.47/33.50

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.50
active(g(b)) → mark(c) 96.47/33.50
active(b) → mark(c) 96.47/33.50
active(g(z0)) → g(active(z0)) 96.47/33.50
g(mark(z0)) → mark(g(z0)) 96.47/33.50
g(ok(z0)) → ok(g(z0)) 96.47/33.50
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.50
proper(g(z0)) → g(proper(z0)) 96.47/33.50
proper(b) → ok(b) 96.47/33.50
proper(c) → ok(c) 96.47/33.50
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.50
top(mark(z0)) → top(proper(z0)) 96.47/33.50
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.50
G(mark(z0)) → c5(G(z0)) 96.47/33.50
G(ok(z0)) → c6(G(z0)) 96.47/33.50
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.47/33.50
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.50
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.50
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.50
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.50
ACTIVE(g(f(z0, g(z0), z1))) → c4(G(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1))) 96.47/33.50
ACTIVE(g(g(b))) → c4(G(mark(c)), ACTIVE(g(b))) 96.47/33.50
ACTIVE(g(b)) → c4(G(mark(c)), ACTIVE(b)) 96.47/33.50
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.50
G(mark(z0)) → c5(G(z0)) 96.47/33.50
G(ok(z0)) → c6(G(z0)) 96.47/33.50
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.47/33.50
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.50
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.50
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.50
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.50
ACTIVE(g(f(z0, g(z0), z1))) → c4(G(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1))) 96.47/33.51
ACTIVE(g(g(b))) → c4(G(mark(c)), ACTIVE(g(b))) 96.47/33.51
ACTIVE(g(b)) → c4(G(mark(c)), ACTIVE(b)) 96.47/33.51
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c5, c6, c7, c8, c11, c12, c13, c4

96.47/33.51
96.47/33.51

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

Removed 1 trailing tuple parts
96.47/33.51
96.47/33.51

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.51
active(g(b)) → mark(c) 96.47/33.51
active(b) → mark(c) 96.47/33.51
active(g(z0)) → g(active(z0)) 96.47/33.51
g(mark(z0)) → mark(g(z0)) 96.47/33.51
g(ok(z0)) → ok(g(z0)) 96.47/33.51
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.51
proper(g(z0)) → g(proper(z0)) 96.47/33.51
proper(b) → ok(b) 96.47/33.51
proper(c) → ok(c) 96.47/33.51
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.51
top(mark(z0)) → top(proper(z0)) 96.47/33.51
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.51
G(mark(z0)) → c5(G(z0)) 96.47/33.51
G(ok(z0)) → c6(G(z0)) 96.47/33.51
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.47/33.51
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.51
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.51
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.51
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c4(G(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1))) 96.47/33.51
ACTIVE(g(g(b))) → c4(G(mark(c)), ACTIVE(g(b))) 96.47/33.51
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.51
ACTIVE(g(b)) → c4(G(mark(c)))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.51
G(mark(z0)) → c5(G(z0)) 96.47/33.51
G(ok(z0)) → c6(G(z0)) 96.47/33.51
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.47/33.51
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.51
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.51
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.51
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c4(G(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1))) 96.47/33.51
ACTIVE(g(g(b))) → c4(G(mark(c)), ACTIVE(g(b))) 96.47/33.51
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.51
ACTIVE(g(b)) → c4(G(mark(c)))
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c5, c6, c7, c8, c11, c12, c13, c4, c4

96.47/33.51
96.47/33.51

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

Split RHS of tuples not part of any SCC
96.47/33.51
96.47/33.51

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.51
active(g(b)) → mark(c) 96.47/33.51
active(b) → mark(c) 96.47/33.51
active(g(z0)) → g(active(z0)) 96.47/33.51
g(mark(z0)) → mark(g(z0)) 96.47/33.51
g(ok(z0)) → ok(g(z0)) 96.47/33.51
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.51
proper(g(z0)) → g(proper(z0)) 96.47/33.51
proper(b) → ok(b) 96.47/33.51
proper(c) → ok(c) 96.47/33.51
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.51
top(mark(z0)) → top(proper(z0)) 96.47/33.51
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.51
G(mark(z0)) → c5(G(z0)) 96.47/33.51
G(ok(z0)) → c6(G(z0)) 96.47/33.51
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.47/33.51
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.51
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.51
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.51
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.51
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.51
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.51
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.51
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.51
G(mark(z0)) → c5(G(z0)) 96.47/33.51
G(ok(z0)) → c6(G(z0)) 96.47/33.51
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.47/33.51
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.51
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.51
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.51
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.51
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.51
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.51
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.51
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c5, c6, c7, c8, c11, c12, c13, c4, c4, c2

96.47/33.51
96.47/33.51

(9) 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(g(b)) → c4(G(mark(c))) 96.47/33.51
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.51
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
We considered the (Usable) Rules:

f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.51
active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.51
active(g(b)) → mark(c) 96.47/33.51
active(b) → mark(c) 96.47/33.51
active(g(z0)) → g(active(z0)) 96.47/33.51
g(mark(z0)) → mark(g(z0)) 96.47/33.51
g(ok(z0)) → ok(g(z0)) 96.47/33.51
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.51
proper(g(z0)) → g(proper(z0)) 96.47/33.51
proper(b) → ok(b) 96.47/33.51
proper(c) → ok(c)
And the Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.51
G(mark(z0)) → c5(G(z0)) 96.47/33.51
G(ok(z0)) → c6(G(z0)) 96.47/33.51
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.47/33.51
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.51
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.51
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.51
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.51
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.51
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.51
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.51
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
The order we found is given by the following interpretation:
Polynomial interpretation : 96.47/33.51

POL(ACTIVE(x1)) = x1    96.47/33.51
POL(F(x1, x2, x3)) = 0    96.47/33.51
POL(G(x1)) = 0    96.47/33.51
POL(PROPER(x1)) = 0    96.47/33.51
POL(TOP(x1)) = [4]x1    96.47/33.51
POL(active(x1)) = 0    96.47/33.51
POL(b) = [1]    96.47/33.51
POL(c) = 0    96.47/33.51
POL(c1(x1)) = x1    96.47/33.51
POL(c11(x1)) = x1    96.47/33.51
POL(c12(x1, x2)) = x1 + x2    96.47/33.51
POL(c13(x1, x2)) = x1 + x2    96.47/33.51
POL(c2(x1)) = x1    96.47/33.51
POL(c4(x1)) = x1    96.47/33.51
POL(c4(x1, x2)) = x1 + x2    96.47/33.51
POL(c5(x1)) = x1    96.47/33.51
POL(c6(x1)) = x1    96.47/33.51
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    96.47/33.51
POL(c8(x1, x2)) = x1 + x2    96.47/33.51
POL(f(x1, x2, x3)) = 0    96.47/33.51
POL(g(x1)) = [2]x1    96.47/33.51
POL(mark(x1)) = x1    96.47/33.51
POL(ok(x1)) = x1    96.47/33.51
POL(proper(x1)) = x1   
96.47/33.51
96.47/33.51

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.51
active(g(b)) → mark(c) 96.47/33.51
active(b) → mark(c) 96.47/33.51
active(g(z0)) → g(active(z0)) 96.47/33.51
g(mark(z0)) → mark(g(z0)) 96.47/33.51
g(ok(z0)) → ok(g(z0)) 96.47/33.51
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.51
proper(g(z0)) → g(proper(z0)) 96.47/33.51
proper(b) → ok(b) 96.47/33.51
proper(c) → ok(c) 96.47/33.51
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.51
top(mark(z0)) → top(proper(z0)) 96.47/33.51
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.51
G(mark(z0)) → c5(G(z0)) 96.47/33.51
G(ok(z0)) → c6(G(z0)) 96.47/33.51
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.47/33.51
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.51
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.51
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.51
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.51
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.51
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.51
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.51
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.51
G(mark(z0)) → c5(G(z0)) 96.47/33.51
G(ok(z0)) → c6(G(z0)) 96.47/33.51
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.47/33.51
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.51
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.51
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.51
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.51
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
K tuples:

ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.51
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.51
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c5, c6, c7, c8, c11, c12, c13, c4, c4, c2

96.47/33.51
96.47/33.51

(11) 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))) 96.47/33.51
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.51
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 96.47/33.51
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 96.47/33.51
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)) 96.47/33.51
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.51
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 96.47/33.51
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 96.47/33.51
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)) 96.47/33.51
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.51
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 96.47/33.51
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
96.47/33.51
96.47/33.51

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.51
active(g(b)) → mark(c) 96.47/33.51
active(b) → mark(c) 96.47/33.51
active(g(z0)) → g(active(z0)) 96.47/33.51
g(mark(z0)) → mark(g(z0)) 96.47/33.51
g(ok(z0)) → ok(g(z0)) 96.47/33.51
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.51
proper(g(z0)) → g(proper(z0)) 96.47/33.51
proper(b) → ok(b) 96.47/33.51
proper(c) → ok(c) 96.47/33.51
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.51
top(mark(z0)) → top(proper(z0)) 96.47/33.51
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.51
G(mark(z0)) → c5(G(z0)) 96.47/33.51
G(ok(z0)) → c6(G(z0)) 96.47/33.51
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.51
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.51
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.51
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.51
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.51
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.51
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.51
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.51
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.51
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))) 96.47/33.51
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.51
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 96.47/33.51
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 96.47/33.51
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)) 96.47/33.51
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.51
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 96.47/33.51
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 96.47/33.51
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)) 96.47/33.51
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.51
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 96.47/33.52
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.52
G(mark(z0)) → c5(G(z0)) 96.47/33.52
G(ok(z0)) → c6(G(z0)) 96.47/33.52
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.52
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.52
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.52
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.52
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.52
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.52
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.52
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))) 96.47/33.52
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.52
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 96.47/33.52
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 96.47/33.52
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)) 96.47/33.52
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.52
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 96.47/33.52
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 96.47/33.52
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)) 96.47/33.52
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.52
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 96.47/33.52
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
K tuples:

ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.52
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.52
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c5, c6, c8, c11, c12, c13, c4, c4, c2, c7

96.47/33.52
96.47/33.52

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

Removed 6 trailing tuple parts
96.47/33.52
96.47/33.52

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.52
active(g(b)) → mark(c) 96.47/33.52
active(b) → mark(c) 96.47/33.52
active(g(z0)) → g(active(z0)) 96.47/33.52
g(mark(z0)) → mark(g(z0)) 96.47/33.52
g(ok(z0)) → ok(g(z0)) 96.47/33.52
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.52
proper(g(z0)) → g(proper(z0)) 96.47/33.52
proper(b) → ok(b) 96.47/33.52
proper(c) → ok(c) 96.47/33.52
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.52
top(mark(z0)) → top(proper(z0)) 96.47/33.52
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.52
G(mark(z0)) → c5(G(z0)) 96.47/33.52
G(ok(z0)) → c6(G(z0)) 96.47/33.52
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.52
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.52
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.52
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.52
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.52
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.52
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.52
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.52
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.52
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.52
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))) 96.47/33.52
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.52
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)) 96.47/33.52
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.52
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)) 96.47/33.52
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.52
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.52
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.52
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.52
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.52
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.52
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.52
G(mark(z0)) → c5(G(z0)) 96.47/33.52
G(ok(z0)) → c6(G(z0)) 96.47/33.54
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 96.47/33.54
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.54
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.54
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.54
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.54
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))) 96.47/33.54
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.54
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)) 96.47/33.54
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.54
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)) 96.47/33.54
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:

ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c5, c6, c8, c11, c12, c13, c4, c4, c2, c7, c7

96.47/33.54
96.47/33.54

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

Use narrowing to replace PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) by

PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
PROPER(g(b)) → c8(G(ok(b)), PROPER(b)) 96.47/33.54
PROPER(g(c)) → c8(G(ok(c)), PROPER(c))
96.47/33.54
96.47/33.54

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.54
active(g(b)) → mark(c) 96.47/33.54
active(b) → mark(c) 96.47/33.54
active(g(z0)) → g(active(z0)) 96.47/33.54
g(mark(z0)) → mark(g(z0)) 96.47/33.54
g(ok(z0)) → ok(g(z0)) 96.47/33.54
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.54
proper(g(z0)) → g(proper(z0)) 96.47/33.54
proper(b) → ok(b) 96.47/33.54
proper(c) → ok(c) 96.47/33.54
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.54
top(mark(z0)) → top(proper(z0)) 96.47/33.54
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.54
G(mark(z0)) → c5(G(z0)) 96.47/33.54
G(ok(z0)) → c6(G(z0)) 96.47/33.54
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.54
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.54
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.54
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.54
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.54
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.54
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))) 96.47/33.54
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.54
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)) 96.47/33.54
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.54
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)) 96.47/33.54
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
PROPER(g(b)) → c8(G(ok(b)), PROPER(b)) 96.47/33.54
PROPER(g(c)) → c8(G(ok(c)), PROPER(c))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.54
G(mark(z0)) → c5(G(z0)) 96.47/33.54
G(ok(z0)) → c6(G(z0)) 96.47/33.54
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.54
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.54
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.54
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.54
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))) 96.47/33.54
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.54
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)) 96.47/33.54
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.54
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)) 96.47/33.54
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
PROPER(g(b)) → c8(G(ok(b)), PROPER(b)) 96.47/33.54
PROPER(g(c)) → c8(G(ok(c)), PROPER(c))
K tuples:

ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c12, c13, c4, c4, c2, c7, c7, c8

96.47/33.54
96.47/33.54

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

Removed 2 trailing tuple parts
96.47/33.54
96.47/33.54

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.54
active(g(b)) → mark(c) 96.47/33.54
active(b) → mark(c) 96.47/33.54
active(g(z0)) → g(active(z0)) 96.47/33.54
g(mark(z0)) → mark(g(z0)) 96.47/33.54
g(ok(z0)) → ok(g(z0)) 96.47/33.54
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.54
proper(g(z0)) → g(proper(z0)) 96.47/33.54
proper(b) → ok(b) 96.47/33.54
proper(c) → ok(c) 96.47/33.54
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.54
top(mark(z0)) → top(proper(z0)) 96.47/33.54
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.54
G(mark(z0)) → c5(G(z0)) 96.47/33.54
G(ok(z0)) → c6(G(z0)) 96.47/33.54
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.54
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.54
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.54
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.54
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.54
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.54
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))) 96.47/33.54
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.54
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)) 96.47/33.54
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.54
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)) 96.47/33.54
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.54
PROPER(g(c)) → c8(G(ok(c)))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.54
G(mark(z0)) → c5(G(z0)) 96.47/33.54
G(ok(z0)) → c6(G(z0)) 96.47/33.54
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.54
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) 96.47/33.54
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.54
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.54
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))) 96.47/33.54
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.54
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)) 96.47/33.54
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.54
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)) 96.47/33.54
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.54
PROPER(g(c)) → c8(G(ok(c)))
K tuples:

ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c12, c13, c4, c4, c2, c7, c7, c8, c8

96.47/33.54
96.47/33.54

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

Use narrowing to replace TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
TOP(mark(b)) → c12(TOP(ok(b)), PROPER(b)) 96.47/33.54
TOP(mark(c)) → c12(TOP(ok(c)), PROPER(c))
96.47/33.54
96.47/33.54

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.54
active(g(b)) → mark(c) 96.47/33.54
active(b) → mark(c) 96.47/33.54
active(g(z0)) → g(active(z0)) 96.47/33.54
g(mark(z0)) → mark(g(z0)) 96.47/33.54
g(ok(z0)) → ok(g(z0)) 96.47/33.54
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.54
proper(g(z0)) → g(proper(z0)) 96.47/33.54
proper(b) → ok(b) 96.47/33.54
proper(c) → ok(c) 96.47/33.54
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.54
top(mark(z0)) → top(proper(z0)) 96.47/33.54
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.54
G(mark(z0)) → c5(G(z0)) 96.47/33.54
G(ok(z0)) → c6(G(z0)) 96.47/33.54
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.54
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.54
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.54
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.54
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.54
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))) 96.47/33.54
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.54
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)) 96.47/33.54
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.54
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)) 96.47/33.54
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.54
PROPER(g(c)) → c8(G(ok(c))) 96.47/33.54
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
TOP(mark(b)) → c12(TOP(ok(b)), PROPER(b)) 96.47/33.54
TOP(mark(c)) → c12(TOP(ok(c)), PROPER(c))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.54
G(mark(z0)) → c5(G(z0)) 96.47/33.54
G(ok(z0)) → c6(G(z0)) 96.47/33.54
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.54
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.54
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.54
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))) 96.47/33.54
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.54
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)) 96.47/33.54
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.54
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)) 96.47/33.54
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.54
PROPER(g(c)) → c8(G(ok(c))) 96.47/33.54
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
TOP(mark(b)) → c12(TOP(ok(b)), PROPER(b)) 96.47/33.54
TOP(mark(c)) → c12(TOP(ok(c)), PROPER(c))
K tuples:

ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c13, c4, c4, c2, c7, c7, c8, c8, c12

96.47/33.54
96.47/33.54

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

Removed 2 trailing tuple parts
96.47/33.54
96.47/33.54

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.54
active(g(b)) → mark(c) 96.47/33.54
active(b) → mark(c) 96.47/33.54
active(g(z0)) → g(active(z0)) 96.47/33.54
g(mark(z0)) → mark(g(z0)) 96.47/33.54
g(ok(z0)) → ok(g(z0)) 96.47/33.54
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.54
proper(g(z0)) → g(proper(z0)) 96.47/33.54
proper(b) → ok(b) 96.47/33.54
proper(c) → ok(c) 96.47/33.54
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.54
top(mark(z0)) → top(proper(z0)) 96.47/33.54
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.54
G(mark(z0)) → c5(G(z0)) 96.47/33.54
G(ok(z0)) → c6(G(z0)) 96.47/33.54
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.54
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.54
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.54
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.54
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.54
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))) 96.47/33.54
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.54
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)) 96.47/33.54
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.54
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)) 96.47/33.54
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.54
PROPER(g(c)) → c8(G(ok(c))) 96.47/33.54
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
TOP(mark(b)) → c12(TOP(ok(b))) 96.47/33.54
TOP(mark(c)) → c12(TOP(ok(c)))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.54
G(mark(z0)) → c5(G(z0)) 96.47/33.54
G(ok(z0)) → c6(G(z0)) 96.47/33.54
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.54
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.54
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.54
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))) 96.47/33.54
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.54
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)) 96.47/33.54
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.54
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)) 96.47/33.54
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.54
PROPER(g(c)) → c8(G(ok(c))) 96.47/33.54
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
TOP(mark(b)) → c12(TOP(ok(b))) 96.47/33.54
TOP(mark(c)) → c12(TOP(ok(c)))
K tuples:

ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c13, c4, c4, c2, c7, c7, c8, c8, c12, c12

96.47/33.54
96.47/33.54

(23) 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)) → c12(TOP(ok(c)))
We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.54
proper(g(z0)) → g(proper(z0)) 96.47/33.54
proper(b) → ok(b) 96.47/33.54
proper(c) → ok(c) 96.47/33.54
g(mark(z0)) → mark(g(z0)) 96.47/33.54
g(ok(z0)) → ok(g(z0)) 96.47/33.54
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.54
active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.54
active(g(b)) → mark(c) 96.47/33.54
active(b) → mark(c) 96.47/33.54
active(g(z0)) → g(active(z0))
And the Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.54
G(mark(z0)) → c5(G(z0)) 96.47/33.54
G(ok(z0)) → c6(G(z0)) 96.47/33.54
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.54
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.54
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.54
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.54
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.54
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))) 96.47/33.54
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.54
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)) 96.47/33.54
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.54
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)) 96.47/33.54
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.54
PROPER(g(c)) → c8(G(ok(c))) 96.47/33.54
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
TOP(mark(b)) → c12(TOP(ok(b))) 96.47/33.54
TOP(mark(c)) → c12(TOP(ok(c)))
The order we found is given by the following interpretation:
Polynomial interpretation : 96.47/33.54

POL(ACTIVE(x1)) = 0    96.47/33.54
POL(F(x1, x2, x3)) = 0    96.47/33.54
POL(G(x1)) = 0    96.47/33.54
POL(PROPER(x1)) = 0    96.47/33.54
POL(TOP(x1)) = x1    96.47/33.54
POL(active(x1)) = x1    96.47/33.54
POL(b) = [1]    96.47/33.54
POL(c) = 0    96.47/33.54
POL(c1(x1)) = x1    96.47/33.54
POL(c11(x1)) = x1    96.47/33.54
POL(c12(x1)) = x1    96.47/33.54
POL(c12(x1, x2)) = x1 + x2    96.47/33.54
POL(c13(x1, x2)) = x1 + x2    96.47/33.54
POL(c2(x1)) = x1    96.47/33.54
POL(c4(x1)) = x1    96.47/33.54
POL(c4(x1, x2)) = x1 + x2    96.47/33.54
POL(c5(x1)) = x1    96.47/33.54
POL(c6(x1)) = x1    96.47/33.54
POL(c7(x1, x2, x3)) = x1 + x2 + x3    96.47/33.54
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    96.47/33.54
POL(c8(x1)) = x1    96.47/33.54
POL(c8(x1, x2)) = x1 + x2    96.47/33.54
POL(f(x1, x2, x3)) = [1]    96.47/33.54
POL(g(x1)) = [1]    96.47/33.54
POL(mark(x1)) = [1]    96.47/33.54
POL(ok(x1)) = x1    96.47/33.54
POL(proper(x1)) = 0   
96.47/33.54
96.47/33.54

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.54
active(g(b)) → mark(c) 96.47/33.54
active(b) → mark(c) 96.47/33.54
active(g(z0)) → g(active(z0)) 96.47/33.54
g(mark(z0)) → mark(g(z0)) 96.47/33.54
g(ok(z0)) → ok(g(z0)) 96.47/33.54
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.54
proper(g(z0)) → g(proper(z0)) 96.47/33.54
proper(b) → ok(b) 96.47/33.54
proper(c) → ok(c) 96.47/33.54
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.54
top(mark(z0)) → top(proper(z0)) 96.47/33.54
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.54
G(mark(z0)) → c5(G(z0)) 96.47/33.54
G(ok(z0)) → c6(G(z0)) 96.47/33.54
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.54
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.54
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.54
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.54
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.54
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))) 96.47/33.54
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.54
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)) 96.47/33.54
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.54
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)) 96.47/33.54
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.54
PROPER(g(c)) → c8(G(ok(c))) 96.47/33.54
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
TOP(mark(b)) → c12(TOP(ok(b))) 96.47/33.54
TOP(mark(c)) → c12(TOP(ok(c)))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.54
G(mark(z0)) → c5(G(z0)) 96.47/33.54
G(ok(z0)) → c6(G(z0)) 96.47/33.54
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.54
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.54
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.54
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))) 96.47/33.54
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.54
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)) 96.47/33.54
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.54
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)) 96.47/33.54
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.54
PROPER(g(c)) → c8(G(ok(c))) 96.47/33.54
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
TOP(mark(b)) → c12(TOP(ok(b)))
K tuples:

ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.54
TOP(mark(c)) → c12(TOP(ok(c)))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c13, c4, c4, c2, c7, c7, c8, c8, c12, c12

96.47/33.54
96.47/33.54

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

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

TOP(mark(b)) → c12(TOP(ok(b)))
We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.54
proper(g(z0)) → g(proper(z0)) 96.47/33.54
proper(b) → ok(b) 96.47/33.54
proper(c) → ok(c) 96.47/33.54
g(mark(z0)) → mark(g(z0)) 96.47/33.54
g(ok(z0)) → ok(g(z0)) 96.47/33.54
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.54
active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.54
active(g(b)) → mark(c) 96.47/33.54
active(b) → mark(c) 96.47/33.54
active(g(z0)) → g(active(z0))
And the Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.54
G(mark(z0)) → c5(G(z0)) 96.47/33.54
G(ok(z0)) → c6(G(z0)) 96.47/33.54
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.54
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.54
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.54
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.54
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.54
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.54
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.54
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))) 96.47/33.54
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.54
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)) 96.47/33.54
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.54
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)) 96.47/33.54
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.54
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.54
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.54
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.54
PROPER(g(c)) → c8(G(ok(c))) 96.47/33.54
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.54
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0))) 96.47/33.54
TOP(mark(b)) → c12(TOP(ok(b))) 96.47/33.54
TOP(mark(c)) → c12(TOP(ok(c)))
The order we found is given by the following interpretation:
Polynomial interpretation : 96.47/33.54

POL(ACTIVE(x1)) = 0    96.47/33.54
POL(F(x1, x2, x3)) = 0    96.47/33.54
POL(G(x1)) = 0    96.47/33.54
POL(PROPER(x1)) = 0    96.47/33.54
POL(TOP(x1)) = x1    96.47/33.54
POL(active(x1)) = 0    96.47/33.54
POL(b) = [1]    96.47/33.54
POL(c) = 0    96.47/33.54
POL(c1(x1)) = x1    96.47/33.54
POL(c11(x1)) = x1    96.47/33.54
POL(c12(x1)) = x1    96.47/33.54
POL(c12(x1, x2)) = x1 + x2    96.47/33.54
POL(c13(x1, x2)) = x1 + x2    96.47/33.54
POL(c2(x1)) = x1    96.47/33.54
POL(c4(x1)) = x1    96.47/33.54
POL(c4(x1, x2)) = x1 + x2    96.47/33.54
POL(c5(x1)) = x1    96.47/33.54
POL(c6(x1)) = x1    96.47/33.54
POL(c7(x1, x2, x3)) = x1 + x2 + x3    96.47/33.54
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    96.47/33.54
POL(c8(x1)) = x1    96.47/33.54
POL(c8(x1, x2)) = x1 + x2    96.47/33.54
POL(f(x1, x2, x3)) = 0    96.47/33.54
POL(g(x1)) = 0    96.47/33.54
POL(mark(x1)) = x1    96.47/33.54
POL(ok(x1)) = 0    96.47/33.54
POL(proper(x1)) = 0   
96.47/33.54
96.47/33.54

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.55
active(g(b)) → mark(c) 96.47/33.55
active(b) → mark(c) 96.47/33.55
active(g(z0)) → g(active(z0)) 96.47/33.55
g(mark(z0)) → mark(g(z0)) 96.47/33.55
g(ok(z0)) → ok(g(z0)) 96.47/33.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.55
proper(g(z0)) → g(proper(z0)) 96.47/33.55
proper(b) → ok(b) 96.47/33.55
proper(c) → ok(c) 96.47/33.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.55
top(mark(z0)) → top(proper(z0)) 96.47/33.55
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.55
G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.55
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.55
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.55
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.55
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.55
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.55
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.55
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.55
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))) 96.47/33.55
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.55
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)) 96.47/33.55
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.55
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)) 96.47/33.55
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.55
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.55
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.55
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.55
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.55
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.55
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.55
PROPER(g(c)) → c8(G(ok(c))) 96.47/33.55
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.55
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0))) 96.47/33.55
TOP(mark(b)) → c12(TOP(ok(b))) 96.47/33.55
TOP(mark(c)) → c12(TOP(ok(c)))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.55
G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.55
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) 96.47/33.55
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.55
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.55
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.55
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))) 96.47/33.55
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.55
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)) 96.47/33.55
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.55
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)) 96.47/33.55
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.55
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.55
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.55
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.55
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.55
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.55
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.55
PROPER(g(c)) → c8(G(ok(c))) 96.47/33.55
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.55
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
K tuples:

ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.55
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.55
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.55
TOP(mark(c)) → c12(TOP(ok(c))) 96.47/33.55
TOP(mark(b)) → c12(TOP(ok(b)))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c13, c4, c4, c2, c7, c7, c8, c8, c12, c12

96.47/33.55
96.47/33.55

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

Use narrowing to replace TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(z0, g(z0), z1))) → c13(TOP(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1))) 96.47/33.55
TOP(ok(g(b))) → c13(TOP(mark(c)), ACTIVE(g(b))) 96.47/33.55
TOP(ok(b)) → c13(TOP(mark(c)), ACTIVE(b)) 96.47/33.55
TOP(ok(g(z0))) → c13(TOP(g(active(z0))), ACTIVE(g(z0)))
96.47/33.55
96.47/33.55

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.55
active(g(b)) → mark(c) 96.47/33.55
active(b) → mark(c) 96.47/33.55
active(g(z0)) → g(active(z0)) 96.47/33.55
g(mark(z0)) → mark(g(z0)) 96.47/33.55
g(ok(z0)) → ok(g(z0)) 96.47/33.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.55
proper(g(z0)) → g(proper(z0)) 96.47/33.55
proper(b) → ok(b) 96.47/33.55
proper(c) → ok(c) 96.47/33.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.55
top(mark(z0)) → top(proper(z0)) 96.47/33.55
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.55
G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.55
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.55
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.55
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.55
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.55
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.55
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.55
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))) 96.47/33.55
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.55
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)) 96.47/33.55
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.55
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)) 96.47/33.55
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.55
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.55
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.55
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.55
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.55
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.55
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.55
PROPER(g(c)) → c8(G(ok(c))) 96.47/33.55
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.55
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0))) 96.47/33.55
TOP(mark(b)) → c12(TOP(ok(b))) 96.47/33.55
TOP(mark(c)) → c12(TOP(ok(c))) 96.47/33.55
TOP(ok(f(z0, g(z0), z1))) → c13(TOP(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1))) 96.47/33.55
TOP(ok(g(b))) → c13(TOP(mark(c)), ACTIVE(g(b))) 96.47/33.55
TOP(ok(b)) → c13(TOP(mark(c)), ACTIVE(b)) 96.47/33.55
TOP(ok(g(z0))) → c13(TOP(g(active(z0))), ACTIVE(g(z0)))
S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.55
G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.55
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.55
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.55
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.55
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))) 96.47/33.55
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.55
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)) 96.47/33.55
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.55
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)) 96.47/33.55
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.55
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.55
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.55
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.55
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.55
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.55
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.55
PROPER(g(c)) → c8(G(ok(c))) 96.47/33.55
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.55
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0))) 96.47/33.55
TOP(ok(f(z0, g(z0), z1))) → c13(TOP(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1))) 96.47/33.55
TOP(ok(g(b))) → c13(TOP(mark(c)), ACTIVE(g(b))) 96.47/33.55
TOP(ok(b)) → c13(TOP(mark(c)), ACTIVE(b)) 96.47/33.55
TOP(ok(g(z0))) → c13(TOP(g(active(z0))), ACTIVE(g(z0)))
K tuples:

ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.55
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.55
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.55
TOP(mark(c)) → c12(TOP(ok(c))) 96.47/33.55
TOP(mark(b)) → c12(TOP(ok(b)))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, F, PROPER, TOP

Compound Symbols:

c1, c5, c6, c11, c4, c4, c2, c7, c7, c8, c8, c12, c12, c13

96.47/33.55
96.47/33.55

(29) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1)) 96.47/33.55
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0))) 96.47/33.55
ACTIVE(g(b)) → c4(G(mark(c))) 96.47/33.55
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1)))) 96.47/33.55
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1))) 96.47/33.55
ACTIVE(g(g(b))) → c2(G(mark(c))) 96.47/33.55
ACTIVE(g(g(b))) → c2(ACTIVE(g(b))) 96.47/33.55
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))) 96.47/33.55
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0))) 96.47/33.55
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)) 96.47/33.55
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2)) 96.47/33.55
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)) 96.47/33.55
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 96.47/33.55
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.47/33.55
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.55
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.47/33.55
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.47/33.55
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.55
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 96.47/33.55
PROPER(g(b)) → c8(G(ok(b))) 96.47/33.55
PROPER(g(c)) → c8(G(ok(c))) 96.47/33.55
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 96.47/33.55
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0))) 96.47/33.55
TOP(ok(f(z0, g(z0), z1))) → c13(TOP(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1))) 96.47/33.55
TOP(ok(g(b))) → c13(TOP(mark(c)), ACTIVE(g(b))) 96.47/33.55
TOP(ok(g(z0))) → c13(TOP(g(active(z0))), ACTIVE(g(z0)))
96.47/33.55
96.47/33.55

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.55
active(g(b)) → mark(c) 96.47/33.55
active(b) → mark(c) 96.47/33.55
active(g(z0)) → g(active(z0)) 96.47/33.55
g(mark(z0)) → mark(g(z0)) 96.47/33.55
g(ok(z0)) → ok(g(z0)) 96.47/33.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.55
proper(g(z0)) → g(proper(z0)) 96.47/33.55
proper(b) → ok(b) 96.47/33.55
proper(c) → ok(c) 96.47/33.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.55
top(mark(z0)) → top(proper(z0)) 96.47/33.55
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.55
TOP(mark(b)) → c12(TOP(ok(b))) 96.47/33.55
TOP(mark(c)) → c12(TOP(ok(c))) 96.47/33.55
TOP(ok(b)) → c13(TOP(mark(c)), ACTIVE(b))
S tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.55
TOP(ok(b)) → c13(TOP(mark(c)), ACTIVE(b))
K tuples:

TOP(mark(c)) → c12(TOP(ok(c))) 96.47/33.55
TOP(mark(b)) → c12(TOP(ok(b)))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, TOP

Compound Symbols:

c5, c6, c11, c12, c13

96.47/33.55
96.47/33.55

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

Removed 2 trailing tuple parts
96.47/33.55
96.47/33.55

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.55
active(g(b)) → mark(c) 96.47/33.55
active(b) → mark(c) 96.47/33.55
active(g(z0)) → g(active(z0)) 96.47/33.55
g(mark(z0)) → mark(g(z0)) 96.47/33.55
g(ok(z0)) → ok(g(z0)) 96.47/33.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.55
proper(g(z0)) → g(proper(z0)) 96.47/33.55
proper(b) → ok(b) 96.47/33.55
proper(c) → ok(c) 96.47/33.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.55
top(mark(z0)) → top(proper(z0)) 96.47/33.55
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.55
TOP(mark(b)) → c12(TOP(ok(b))) 96.47/33.55
TOP(mark(c)) → c12 96.47/33.55
TOP(ok(b)) → c13(TOP(mark(c)))
S tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2)) 96.47/33.55
TOP(ok(b)) → c13(TOP(mark(c)))
K tuples:

TOP(mark(b)) → c12(TOP(ok(b))) 96.47/33.55
TOP(mark(c)) → c12
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F, TOP

Compound Symbols:

c5, c6, c11, c12, c12, c13

96.47/33.55
96.47/33.55

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

Removed 3 trailing nodes:

TOP(mark(b)) → c12(TOP(ok(b))) 96.47/33.55
TOP(mark(c)) → c12 96.47/33.55
TOP(ok(b)) → c13(TOP(mark(c)))
96.47/33.55
96.47/33.55

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.55
active(g(b)) → mark(c) 96.47/33.55
active(b) → mark(c) 96.47/33.55
active(g(z0)) → g(active(z0)) 96.47/33.55
g(mark(z0)) → mark(g(z0)) 96.47/33.55
g(ok(z0)) → ok(g(z0)) 96.47/33.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.55
proper(g(z0)) → g(proper(z0)) 96.47/33.55
proper(b) → ok(b) 96.47/33.55
proper(c) → ok(c) 96.47/33.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.55
top(mark(z0)) → top(proper(z0)) 96.47/33.55
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
S tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F

Compound Symbols:

c5, c6, c11

96.47/33.55
96.47/33.55

(35) 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.

G(mark(z0)) → c5(G(z0))
We considered the (Usable) Rules:none
And the Tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 96.47/33.55

POL(F(x1, x2, x3)) = x1 + x2    96.47/33.55
POL(G(x1)) = [2]x1    96.47/33.55
POL(c11(x1)) = x1    96.47/33.55
POL(c5(x1)) = x1    96.47/33.55
POL(c6(x1)) = x1    96.47/33.55
POL(mark(x1)) = [3] + x1    96.47/33.55
POL(ok(x1)) = x1   
96.47/33.55
96.47/33.55

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.55
active(g(b)) → mark(c) 96.47/33.55
active(b) → mark(c) 96.47/33.55
active(g(z0)) → g(active(z0)) 96.47/33.55
g(mark(z0)) → mark(g(z0)) 96.47/33.55
g(ok(z0)) → ok(g(z0)) 96.47/33.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.55
proper(g(z0)) → g(proper(z0)) 96.47/33.55
proper(b) → ok(b) 96.47/33.55
proper(c) → ok(c) 96.47/33.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.55
top(mark(z0)) → top(proper(z0)) 96.47/33.55
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
S tuples:

G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
K tuples:

G(mark(z0)) → c5(G(z0))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F

Compound Symbols:

c5, c6, c11

96.47/33.55
96.47/33.55

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

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

G(ok(z0)) → c6(G(z0))
We considered the (Usable) Rules:none
And the Tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 96.47/33.55

POL(F(x1, x2, x3)) = 0    96.47/33.55
POL(G(x1)) = [2]x1    96.47/33.55
POL(c11(x1)) = x1    96.47/33.55
POL(c5(x1)) = x1    96.47/33.55
POL(c6(x1)) = x1    96.47/33.55
POL(mark(x1)) = x1    96.47/33.55
POL(ok(x1)) = [1] + x1   
96.47/33.55
96.47/33.55

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.55
active(g(b)) → mark(c) 96.47/33.55
active(b) → mark(c) 96.47/33.55
active(g(z0)) → g(active(z0)) 96.47/33.55
g(mark(z0)) → mark(g(z0)) 96.47/33.55
g(ok(z0)) → ok(g(z0)) 96.47/33.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.55
proper(g(z0)) → g(proper(z0)) 96.47/33.55
proper(b) → ok(b) 96.47/33.55
proper(c) → ok(c) 96.47/33.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.55
top(mark(z0)) → top(proper(z0)) 96.47/33.55
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
S tuples:

F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
K tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F

Compound Symbols:

c5, c6, c11

96.47/33.55
96.47/33.55

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

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

F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 96.47/33.55

POL(F(x1, x2, x3)) = x3    96.47/33.55
POL(G(x1)) = [3]x1    96.47/33.55
POL(c11(x1)) = x1    96.47/33.55
POL(c5(x1)) = x1    96.47/33.55
POL(c6(x1)) = x1    96.47/33.55
POL(mark(x1)) = x1    96.47/33.55
POL(ok(x1)) = [2] + x1   
96.47/33.55
96.47/33.55

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1)) 96.47/33.55
active(g(b)) → mark(c) 96.47/33.55
active(b) → mark(c) 96.47/33.55
active(g(z0)) → g(active(z0)) 96.47/33.55
g(mark(z0)) → mark(g(z0)) 96.47/33.55
g(ok(z0)) → ok(g(z0)) 96.47/33.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 96.47/33.55
proper(g(z0)) → g(proper(z0)) 96.47/33.55
proper(b) → ok(b) 96.47/33.55
proper(c) → ok(c) 96.47/33.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 96.47/33.55
top(mark(z0)) → top(proper(z0)) 96.47/33.55
top(ok(z0)) → top(active(z0))
Tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
S tuples:none
K tuples:

G(mark(z0)) → c5(G(z0)) 96.47/33.55
G(ok(z0)) → c6(G(z0)) 96.47/33.55
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

G, F

Compound Symbols:

c5, c6, c11

96.47/33.55
96.47/33.55

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

The set S is empty
96.47/33.55
96.47/33.55

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

96.47/33.55
96.47/33.55
96.71/34.15 EOF