YES(O(1), O(n^1)) 99.78/33.07 YES(O(1), O(n^1)) 99.78/33.09 99.78/33.09 99.78/33.09 99.78/33.09 99.78/33.09 99.78/33.09 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 99.78/33.09 99.78/33.09 99.78/33.09
99.78/33.09
99.78/33.09
99.78/33.09
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
99.78/33.09 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳2 CdtProblem
99.78/33.09 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳4 CdtProblem
99.78/33.09 
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳6 CdtProblem
99.78/33.09 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
99.78/33.09 
↳8 CdtProblem
99.78/33.09 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
99.78/33.09 
↳10 CdtProblem
99.78/33.09 
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳12 CdtProblem
99.78/33.09 
↳13 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳14 CdtProblem
99.78/33.09 
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
99.78/33.09 
↳16 CdtProblem
99.78/33.09 
↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳18 CdtProblem
99.78/33.09 
↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳20 CdtProblem
99.78/33.09 
↳21 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳22 CdtProblem
99.78/33.09 
↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
99.78/33.09 
↳24 CdtProblem
99.78/33.09 
↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳26 CdtProblem
99.78/33.09 
↳27 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳28 CdtProblem
99.78/33.09 
↳29 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳30 CdtProblem
99.78/33.09 
↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
99.78/33.09 
↳32 CdtProblem
99.78/33.09 
↳33 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳34 CdtProblem
99.78/33.09 
↳35 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳36 CdtProblem
99.78/33.09 
↳37 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳38 CdtProblem
99.78/33.09 
↳39 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳40 CdtProblem
99.78/33.09 
↳41 CdtUnreachableProof (⇔)
99.78/33.09 
↳42 CdtProblem
99.78/33.09 
↳43 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳44 CdtProblem
99.78/33.09 
↳45 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳46 CdtProblem
99.78/33.09 
↳47 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
99.78/33.09 
↳48 CdtProblem
99.78/33.09 
↳49 SIsEmptyProof (BOTH BOUNDS(ID, ID))
99.78/33.09 
↳50 BOUNDS(O(1), O(1))
99.78/33.09
99.78/33.09 99.78/33.09
99.78/33.09
99.78/33.09

(0) Obligation:

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

active(f(f(a))) → mark(f(g(f(a)))) 99.78/33.09
active(f(X)) → f(active(X)) 99.78/33.09
f(mark(X)) → mark(f(X)) 99.78/33.09
proper(f(X)) → f(proper(X)) 99.78/33.09
proper(a) → ok(a) 99.78/33.09
proper(g(X)) → g(proper(X)) 99.78/33.09
f(ok(X)) → ok(f(X)) 99.78/33.09
g(ok(X)) → ok(g(X)) 99.78/33.09
top(mark(X)) → top(proper(X)) 99.78/33.09
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
99.78/33.09
99.78/33.09

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

Converted CpxTRS to CDT
99.78/33.09
99.78/33.09

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 99.78/33.09
active(f(z0)) → f(active(z0)) 99.78/33.09
f(mark(z0)) → mark(f(z0)) 99.78/33.09
f(ok(z0)) → ok(f(z0)) 99.78/33.09
proper(f(z0)) → f(proper(z0)) 99.78/33.09
proper(a) → ok(a) 99.78/33.09
proper(g(z0)) → g(proper(z0)) 99.78/33.09
g(ok(z0)) → ok(g(z0)) 99.78/33.09
top(mark(z0)) → top(proper(z0)) 99.78/33.09
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(f(a))) → c(F(g(f(a))), G(f(a)), F(a)) 99.78/33.09
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 99.78/33.09
F(mark(z0)) → c2(F(z0)) 99.78/33.09
F(ok(z0)) → c3(F(z0)) 99.78/33.09
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 99.78/33.09
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 99.78/33.09
G(ok(z0)) → c7(G(z0)) 99.78/33.09
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 99.78/33.09
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(f(a))) → c(F(g(f(a))), G(f(a)), F(a)) 99.78/33.09
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 99.78/33.09
F(mark(z0)) → c2(F(z0)) 99.78/33.09
F(ok(z0)) → c3(F(z0)) 99.78/33.09
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 99.78/33.09
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 99.78/33.09
G(ok(z0)) → c7(G(z0)) 99.78/33.09
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 99.78/33.09
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

ACTIVE, F, PROPER, G, TOP

Compound Symbols:

c, c1, c2, c3, c4, c6, c7, c8, c9

100.00/33.12
100.00/33.12

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

Removed 3 trailing tuple parts
100.00/33.12
100.00/33.12

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.12
active(f(z0)) → f(active(z0)) 100.00/33.12
f(mark(z0)) → mark(f(z0)) 100.00/33.12
f(ok(z0)) → ok(f(z0)) 100.00/33.12
proper(f(z0)) → f(proper(z0)) 100.00/33.12
proper(a) → ok(a) 100.00/33.12
proper(g(z0)) → g(proper(z0)) 100.00/33.12
g(ok(z0)) → ok(g(z0)) 100.00/33.12
top(mark(z0)) → top(proper(z0)) 100.00/33.12
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 100.00/33.12
F(mark(z0)) → c2(F(z0)) 100.00/33.12
F(ok(z0)) → c3(F(z0)) 100.00/33.12
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.12
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.12
G(ok(z0)) → c7(G(z0)) 100.00/33.12
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.12
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.12
ACTIVE(f(f(a))) → c
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 100.00/33.12
F(mark(z0)) → c2(F(z0)) 100.00/33.12
F(ok(z0)) → c3(F(z0)) 100.00/33.12
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.12
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.12
G(ok(z0)) → c7(G(z0)) 100.00/33.12
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.12
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.12
ACTIVE(f(f(a))) → c
K tuples:none
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

ACTIVE, F, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c6, c7, c8, c9, c

100.00/33.12
100.00/33.12

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

Removed 1 trailing nodes:

ACTIVE(f(f(a))) → c
100.00/33.12
100.00/33.12

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.12
active(f(z0)) → f(active(z0)) 100.00/33.12
f(mark(z0)) → mark(f(z0)) 100.00/33.12
f(ok(z0)) → ok(f(z0)) 100.00/33.12
proper(f(z0)) → f(proper(z0)) 100.00/33.12
proper(a) → ok(a) 100.00/33.12
proper(g(z0)) → g(proper(z0)) 100.00/33.12
g(ok(z0)) → ok(g(z0)) 100.00/33.12
top(mark(z0)) → top(proper(z0)) 100.00/33.12
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 100.00/33.12
F(mark(z0)) → c2(F(z0)) 100.00/33.12
F(ok(z0)) → c3(F(z0)) 100.00/33.12
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.12
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.12
G(ok(z0)) → c7(G(z0)) 100.00/33.12
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.12
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.12
ACTIVE(f(f(a))) → c
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 100.00/33.12
F(mark(z0)) → c2(F(z0)) 100.00/33.12
F(ok(z0)) → c3(F(z0)) 100.00/33.12
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.12
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.12
G(ok(z0)) → c7(G(z0)) 100.00/33.12
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.12
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.12
ACTIVE(f(f(a))) → c
K tuples:none
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

ACTIVE, F, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c6, c7, c8, c9, c

100.00/33.12
100.00/33.12

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

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

ACTIVE(f(f(a))) → c
We considered the (Usable) Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.12
active(f(z0)) → f(active(z0)) 100.00/33.12
f(mark(z0)) → mark(f(z0)) 100.00/33.12
f(ok(z0)) → ok(f(z0)) 100.00/33.12
proper(f(z0)) → f(proper(z0)) 100.00/33.12
proper(a) → ok(a) 100.00/33.12
proper(g(z0)) → g(proper(z0)) 100.00/33.12
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 100.00/33.12
F(mark(z0)) → c2(F(z0)) 100.00/33.12
F(ok(z0)) → c3(F(z0)) 100.00/33.12
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.12
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.12
G(ok(z0)) → c7(G(z0)) 100.00/33.12
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.12
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.12
ACTIVE(f(f(a))) → c
The order we found is given by the following interpretation:
Polynomial interpretation : 100.00/33.12

POL(ACTIVE(x1)) = x1    100.00/33.12
POL(F(x1)) = 0    100.00/33.12
POL(G(x1)) = 0    100.00/33.12
POL(PROPER(x1)) = 0    100.00/33.12
POL(TOP(x1)) = x1    100.00/33.12
POL(a) = [1]    100.00/33.12
POL(active(x1)) = 0    100.00/33.12
POL(c) = 0    100.00/33.12
POL(c1(x1, x2)) = x1 + x2    100.00/33.12
POL(c2(x1)) = x1    100.00/33.12
POL(c3(x1)) = x1    100.00/33.12
POL(c4(x1, x2)) = x1 + x2    100.00/33.12
POL(c6(x1, x2)) = x1 + x2    100.00/33.12
POL(c7(x1)) = x1    100.00/33.12
POL(c8(x1, x2)) = x1 + x2    100.00/33.12
POL(c9(x1, x2)) = x1 + x2    100.00/33.12
POL(f(x1)) = [4]x1    100.00/33.12
POL(g(x1)) = 0    100.00/33.12
POL(mark(x1)) = x1    100.00/33.12
POL(ok(x1)) = x1    100.00/33.12
POL(proper(x1)) = x1   
100.00/33.12
100.00/33.12

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.12
active(f(z0)) → f(active(z0)) 100.00/33.12
f(mark(z0)) → mark(f(z0)) 100.00/33.12
f(ok(z0)) → ok(f(z0)) 100.00/33.12
proper(f(z0)) → f(proper(z0)) 100.00/33.12
proper(a) → ok(a) 100.00/33.12
proper(g(z0)) → g(proper(z0)) 100.00/33.12
g(ok(z0)) → ok(g(z0)) 100.00/33.12
top(mark(z0)) → top(proper(z0)) 100.00/33.12
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 100.00/33.12
F(mark(z0)) → c2(F(z0)) 100.00/33.12
F(ok(z0)) → c3(F(z0)) 100.00/33.12
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.12
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.12
G(ok(z0)) → c7(G(z0)) 100.00/33.12
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.12
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.12
ACTIVE(f(f(a))) → c
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 100.00/33.12
F(mark(z0)) → c2(F(z0)) 100.00/33.12
F(ok(z0)) → c3(F(z0)) 100.00/33.12
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.12
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.12
G(ok(z0)) → c7(G(z0)) 100.00/33.12
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.12
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
K tuples:

ACTIVE(f(f(a))) → c
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

ACTIVE, F, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c6, c7, c8, c9, c

100.00/33.12
100.00/33.12

(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.

TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
We considered the (Usable) Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.12
active(f(z0)) → f(active(z0)) 100.00/33.12
f(mark(z0)) → mark(f(z0)) 100.00/33.12
f(ok(z0)) → ok(f(z0)) 100.00/33.12
proper(f(z0)) → f(proper(z0)) 100.00/33.12
proper(a) → ok(a) 100.00/33.12
proper(g(z0)) → g(proper(z0)) 100.00/33.12
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 100.00/33.12
F(mark(z0)) → c2(F(z0)) 100.00/33.12
F(ok(z0)) → c3(F(z0)) 100.00/33.12
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.12
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.12
G(ok(z0)) → c7(G(z0)) 100.00/33.12
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.12
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.12
ACTIVE(f(f(a))) → c
The order we found is given by the following interpretation:
Polynomial interpretation : 100.00/33.12

POL(ACTIVE(x1)) = 0    100.00/33.12
POL(F(x1)) = 0    100.00/33.12
POL(G(x1)) = 0    100.00/33.12
POL(PROPER(x1)) = 0    100.00/33.12
POL(TOP(x1)) = [2]x1    100.00/33.12
POL(a) = [1]    100.00/33.12
POL(active(x1)) = x1    100.00/33.12
POL(c) = 0    100.00/33.12
POL(c1(x1, x2)) = x1 + x2    100.00/33.12
POL(c2(x1)) = x1    100.00/33.12
POL(c3(x1)) = x1    100.00/33.12
POL(c4(x1, x2)) = x1 + x2    100.00/33.12
POL(c6(x1, x2)) = x1 + x2    100.00/33.12
POL(c7(x1)) = x1    100.00/33.12
POL(c8(x1, x2)) = x1 + x2    100.00/33.12
POL(c9(x1, x2)) = x1 + x2    100.00/33.12
POL(f(x1)) = x1    100.00/33.12
POL(g(x1)) = 0    100.00/33.12
POL(mark(x1)) = [1] + x1    100.00/33.12
POL(ok(x1)) = x1    100.00/33.12
POL(proper(x1)) = x1   
100.00/33.12
100.00/33.12

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.12
active(f(z0)) → f(active(z0)) 100.00/33.12
f(mark(z0)) → mark(f(z0)) 100.00/33.12
f(ok(z0)) → ok(f(z0)) 100.00/33.12
proper(f(z0)) → f(proper(z0)) 100.00/33.13
proper(a) → ok(a) 100.00/33.13
proper(g(z0)) → g(proper(z0)) 100.00/33.13
g(ok(z0)) → ok(g(z0)) 100.00/33.13
top(mark(z0)) → top(proper(z0)) 100.00/33.13
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 100.00/33.13
F(mark(z0)) → c2(F(z0)) 100.00/33.13
F(ok(z0)) → c3(F(z0)) 100.00/33.13
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.13
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.13
G(ok(z0)) → c7(G(z0)) 100.00/33.13
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.13
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.13
ACTIVE(f(f(a))) → c
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 100.00/33.13
F(mark(z0)) → c2(F(z0)) 100.00/33.13
F(ok(z0)) → c3(F(z0)) 100.00/33.13
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.13
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.13
G(ok(z0)) → c7(G(z0)) 100.00/33.13
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.13
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

ACTIVE, F, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c6, c7, c8, c9, c

100.00/33.13
100.00/33.13

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

Use narrowing to replace ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) by

ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.13
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
100.00/33.13
100.00/33.13

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.13
active(f(z0)) → f(active(z0)) 100.00/33.13
f(mark(z0)) → mark(f(z0)) 100.00/33.13
f(ok(z0)) → ok(f(z0)) 100.00/33.13
proper(f(z0)) → f(proper(z0)) 100.00/33.13
proper(a) → ok(a) 100.00/33.13
proper(g(z0)) → g(proper(z0)) 100.00/33.13
g(ok(z0)) → ok(g(z0)) 100.00/33.13
top(mark(z0)) → top(proper(z0)) 100.00/33.13
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.13
F(ok(z0)) → c3(F(z0)) 100.00/33.13
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.13
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.13
G(ok(z0)) → c7(G(z0)) 100.00/33.13
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.13
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.13
ACTIVE(f(f(a))) → c 100.00/33.13
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.13
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.13
F(ok(z0)) → c3(F(z0)) 100.00/33.13
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, PROPER, G, TOP, ACTIVE

Compound Symbols:

c2, c3, c4, c6, c7, c8, c9, c, c1

100.00/33.14
100.00/33.14

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

Removed 1 trailing nodes:

ACTIVE(f(f(a))) → c
100.00/33.14
100.00/33.14

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, PROPER, G, TOP, ACTIVE

Compound Symbols:

c2, c3, c4, c6, c7, c8, c9, c, c1

100.00/33.14
100.00/33.14

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

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

ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
We considered the (Usable) Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0))
And the Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation : 100.00/33.14

POL(ACTIVE(x1)) = [2]x1    100.00/33.14
POL(F(x1)) = 0    100.00/33.14
POL(G(x1)) = 0    100.00/33.14
POL(PROPER(x1)) = 0    100.00/33.14
POL(TOP(x1)) = [2]x1    100.00/33.14
POL(a) = [4]    100.00/33.14
POL(active(x1)) = 0    100.00/33.14
POL(c) = 0    100.00/33.14
POL(c1(x1, x2)) = x1 + x2    100.00/33.14
POL(c2(x1)) = x1    100.00/33.14
POL(c3(x1)) = x1    100.00/33.14
POL(c4(x1, x2)) = x1 + x2    100.00/33.14
POL(c6(x1, x2)) = x1 + x2    100.00/33.14
POL(c7(x1)) = x1    100.00/33.14
POL(c8(x1, x2)) = x1 + x2    100.00/33.14
POL(c9(x1, x2)) = x1 + x2    100.00/33.14
POL(f(x1)) = [4]x1    100.00/33.14
POL(g(x1)) = 0    100.00/33.14
POL(mark(x1)) = x1    100.00/33.14
POL(ok(x1)) = x1    100.00/33.14
POL(proper(x1)) = x1   
100.00/33.14
100.00/33.14

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, PROPER, G, TOP, ACTIVE

Compound Symbols:

c2, c3, c4, c6, c7, c8, c9, c, c1

100.00/33.14
100.00/33.14

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

Use narrowing to replace PROPER(f(z0)) → c4(F(proper(z0)), PROPER(z0)) by

PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a)), PROPER(a)) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
100.00/33.14
100.00/33.14

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a)), PROPER(a)) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a)), PROPER(a)) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, PROPER, G, TOP, ACTIVE

Compound Symbols:

c2, c3, c6, c7, c8, c9, c, c1, c4

100.00/33.14
100.00/33.14

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

Removed 1 trailing tuple parts
100.00/33.14
100.00/33.14

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, PROPER, G, TOP, ACTIVE

Compound Symbols:

c2, c3, c6, c7, c8, c9, c, c1, c4, c4

100.00/33.14
100.00/33.14

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

Removed 1 trailing nodes:

ACTIVE(f(f(a))) → c
100.00/33.14
100.00/33.14

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a))))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, PROPER, G, TOP, ACTIVE

Compound Symbols:

c2, c3, c6, c7, c8, c9, c, c1, c4, c4

100.00/33.14
100.00/33.14

(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.

PROPER(f(a)) → c4(F(ok(a)))
We considered the (Usable) Rules:

proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0))
And the Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a)))
The order we found is given by the following interpretation:
Polynomial interpretation : 100.00/33.14

POL(ACTIVE(x1)) = 0    100.00/33.14
POL(F(x1)) = 0    100.00/33.14
POL(G(x1)) = 0    100.00/33.14
POL(PROPER(x1)) = [1]    100.00/33.14
POL(TOP(x1)) = x1    100.00/33.14
POL(a) = 0    100.00/33.14
POL(active(x1)) = x1    100.00/33.14
POL(c) = 0    100.00/33.14
POL(c1(x1, x2)) = x1 + x2    100.00/33.14
POL(c2(x1)) = x1    100.00/33.14
POL(c3(x1)) = x1    100.00/33.14
POL(c4(x1)) = x1    100.00/33.14
POL(c4(x1, x2)) = x1 + x2    100.00/33.14
POL(c6(x1, x2)) = x1 + x2    100.00/33.14
POL(c7(x1)) = x1    100.00/33.14
POL(c8(x1, x2)) = x1 + x2    100.00/33.14
POL(c9(x1, x2)) = x1 + x2    100.00/33.14
POL(f(x1)) = [4] + x1    100.00/33.14
POL(g(x1)) = 0    100.00/33.14
POL(mark(x1)) = [4] + x1    100.00/33.14
POL(ok(x1)) = x1    100.00/33.14
POL(proper(x1)) = x1   
100.00/33.14
100.00/33.14

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a)))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, PROPER, G, TOP, ACTIVE

Compound Symbols:

c2, c3, c6, c7, c8, c9, c, c1, c4, c4

100.00/33.14
100.00/33.14

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

Use narrowing to replace PROPER(g(z0)) → c6(G(proper(z0)), PROPER(z0)) by

PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)), PROPER(a)) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
100.00/33.14
100.00/33.14

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)), PROPER(a)) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)), PROPER(a)) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a)))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

c2, c3, c7, c8, c9, c, c1, c4, c4, c6

100.00/33.14
100.00/33.14

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

Removed 1 trailing tuple parts
100.00/33.14
100.00/33.14

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a)))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

c2, c3, c7, c8, c9, c, c1, c4, c4, c6, c6

100.00/33.14
100.00/33.14

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

Removed 1 trailing nodes:

ACTIVE(f(f(a))) → c
100.00/33.14
100.00/33.14

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a)))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

c2, c3, c7, c8, c9, c, c1, c4, c4, c6, c6

100.00/33.14
100.00/33.14

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

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

PROPER(g(a)) → c6(G(ok(a)))
We considered the (Usable) Rules:

proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0))
And the Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)))
The order we found is given by the following interpretation:
Polynomial interpretation : 100.00/33.14

POL(ACTIVE(x1)) = 0    100.00/33.14
POL(F(x1)) = 0    100.00/33.14
POL(G(x1)) = 0    100.00/33.14
POL(PROPER(x1)) = [1]    100.00/33.14
POL(TOP(x1)) = [2]x1    100.00/33.14
POL(a) = [1]    100.00/33.14
POL(active(x1)) = x1    100.00/33.14
POL(c) = 0    100.00/33.14
POL(c1(x1, x2)) = x1 + x2    100.00/33.14
POL(c2(x1)) = x1    100.00/33.14
POL(c3(x1)) = x1    100.00/33.14
POL(c4(x1)) = x1    100.00/33.14
POL(c4(x1, x2)) = x1 + x2    100.00/33.14
POL(c6(x1)) = x1    100.00/33.14
POL(c6(x1, x2)) = x1 + x2    100.00/33.14
POL(c7(x1)) = x1    100.00/33.14
POL(c8(x1, x2)) = x1 + x2    100.00/33.14
POL(c9(x1, x2)) = x1 + x2    100.00/33.14
POL(f(x1)) = [4]x1    100.00/33.14
POL(g(x1)) = 0    100.00/33.14
POL(mark(x1)) = [2] + x1    100.00/33.14
POL(ok(x1)) = x1    100.00/33.14
POL(proper(x1)) = x1   
100.00/33.14
100.00/33.14

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

c2, c3, c7, c8, c9, c, c1, c4, c4, c6, c6

100.00/33.14
100.00/33.14

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

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

TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a)) 100.00/33.14
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
100.00/33.14
100.00/33.14

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a))) 100.00/33.14
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a)) 100.00/33.14
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

c2, c3, c7, c9, c, c1, c4, c4, c6, c6, c8

100.00/33.14
100.00/33.14

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

Removed 1 trailing tuple parts
100.00/33.14
100.00/33.14

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a))) 100.00/33.14
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
TOP(mark(a)) → c8(TOP(ok(a)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

c2, c3, c7, c9, c, c1, c4, c4, c6, c6, c8, c8

100.00/33.14
100.00/33.14

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

Removed 1 trailing nodes:

ACTIVE(f(f(a))) → c
100.00/33.14
100.00/33.14

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a))) 100.00/33.14
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
TOP(mark(a)) → c8(TOP(ok(a)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

c2, c3, c7, c9, c, c1, c4, c4, c6, c6, c8, c8

100.00/33.14
100.00/33.14

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

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

TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
TOP(ok(f(z0))) → c9(TOP(f(active(z0))), ACTIVE(f(z0)))
100.00/33.14
100.00/33.14

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.14
active(f(z0)) → f(active(z0)) 100.00/33.14
f(mark(z0)) → mark(f(z0)) 100.00/33.14
f(ok(z0)) → ok(f(z0)) 100.00/33.14
proper(f(z0)) → f(proper(z0)) 100.00/33.14
proper(a) → ok(a) 100.00/33.14
proper(g(z0)) → g(proper(z0)) 100.00/33.14
g(ok(z0)) → ok(g(z0)) 100.00/33.14
top(mark(z0)) → top(proper(z0)) 100.00/33.14
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a))) 100.00/33.14
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
TOP(mark(a)) → c8(TOP(ok(a))) 100.00/33.14
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
TOP(ok(f(z0))) → c9(TOP(f(active(z0))), ACTIVE(f(z0)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.14
F(ok(z0)) → c3(F(z0)) 100.00/33.14
G(ok(z0)) → c7(G(z0)) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
TOP(ok(f(z0))) → c9(TOP(f(active(z0))), ACTIVE(f(z0)))
K tuples:

ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(a)) → c6(G(ok(a)))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, G, ACTIVE, PROPER, TOP

Compound Symbols:

c2, c3, c7, c, c1, c4, c4, c6, c6, c8, c8, c9

100.00/33.14
100.00/33.14

(41) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(f(f(a))) → c 100.00/33.14
ACTIVE(f(f(f(a)))) → c1(F(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.14
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 100.00/33.14
PROPER(f(f(z0))) → c4(F(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(f(g(z0))) → c4(F(g(proper(z0))), PROPER(g(z0))) 100.00/33.14
PROPER(f(a)) → c4(F(ok(a))) 100.00/33.14
PROPER(g(f(z0))) → c6(G(f(proper(z0))), PROPER(f(z0))) 100.00/33.14
PROPER(g(g(z0))) → c6(G(g(proper(z0))), PROPER(g(z0))) 100.00/33.15
PROPER(g(a)) → c6(G(ok(a))) 100.00/33.15
TOP(mark(f(z0))) → c8(TOP(f(proper(z0))), PROPER(f(z0))) 100.00/33.15
TOP(mark(g(z0))) → c8(TOP(g(proper(z0))), PROPER(g(z0))) 100.00/33.15
TOP(ok(f(f(a)))) → c9(TOP(mark(f(g(f(a))))), ACTIVE(f(f(a)))) 100.00/33.15
TOP(ok(f(z0))) → c9(TOP(f(active(z0))), ACTIVE(f(z0)))
100.00/33.15
100.00/33.15

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.15
active(f(z0)) → f(active(z0)) 100.00/33.15
f(mark(z0)) → mark(f(z0)) 100.00/33.15
f(ok(z0)) → ok(f(z0)) 100.00/33.15
proper(f(z0)) → f(proper(z0)) 100.00/33.15
proper(a) → ok(a) 100.00/33.15
proper(g(z0)) → g(proper(z0)) 100.00/33.15
g(ok(z0)) → ok(g(z0)) 100.00/33.15
top(mark(z0)) → top(proper(z0)) 100.00/33.15
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.15
F(ok(z0)) → c3(F(z0)) 100.00/33.15
G(ok(z0)) → c7(G(z0)) 100.00/33.15
TOP(mark(a)) → c8(TOP(ok(a)))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.15
F(ok(z0)) → c3(F(z0)) 100.00/33.15
G(ok(z0)) → c7(G(z0))
K tuples:none
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, G, TOP

Compound Symbols:

c2, c3, c7, c8

100.00/33.15
100.00/33.15

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

Removed 1 trailing tuple parts
100.00/33.15
100.00/33.15

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.15
active(f(z0)) → f(active(z0)) 100.00/33.15
f(mark(z0)) → mark(f(z0)) 100.00/33.15
f(ok(z0)) → ok(f(z0)) 100.00/33.15
proper(f(z0)) → f(proper(z0)) 100.00/33.15
proper(a) → ok(a) 100.00/33.15
proper(g(z0)) → g(proper(z0)) 100.00/33.15
g(ok(z0)) → ok(g(z0)) 100.00/33.15
top(mark(z0)) → top(proper(z0)) 100.00/33.15
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.15
F(ok(z0)) → c3(F(z0)) 100.00/33.15
G(ok(z0)) → c7(G(z0)) 100.00/33.15
TOP(mark(a)) → c8
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.15
F(ok(z0)) → c3(F(z0)) 100.00/33.15
G(ok(z0)) → c7(G(z0))
K tuples:none
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, G, TOP

Compound Symbols:

c2, c3, c7, c8

100.00/33.15
100.00/33.15

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

Removed 1 trailing nodes:

TOP(mark(a)) → c8
100.00/33.15
100.00/33.15

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.15
active(f(z0)) → f(active(z0)) 100.00/33.15
f(mark(z0)) → mark(f(z0)) 100.00/33.15
f(ok(z0)) → ok(f(z0)) 100.00/33.15
proper(f(z0)) → f(proper(z0)) 100.00/33.15
proper(a) → ok(a) 100.00/33.15
proper(g(z0)) → g(proper(z0)) 100.00/33.15
g(ok(z0)) → ok(g(z0)) 100.00/33.15
top(mark(z0)) → top(proper(z0)) 100.00/33.15
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.15
F(ok(z0)) → c3(F(z0)) 100.00/33.15
G(ok(z0)) → c7(G(z0))
S tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.15
F(ok(z0)) → c3(F(z0)) 100.00/33.15
G(ok(z0)) → c7(G(z0))
K tuples:none
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, G

Compound Symbols:

c2, c3, c7

100.00/33.15
100.00/33.15

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

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

F(mark(z0)) → c2(F(z0)) 100.00/33.15
F(ok(z0)) → c3(F(z0)) 100.00/33.15
G(ok(z0)) → c7(G(z0))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.15
F(ok(z0)) → c3(F(z0)) 100.00/33.15
G(ok(z0)) → c7(G(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 100.00/33.15

POL(F(x1)) = [3]x1    100.00/33.15
POL(G(x1)) = [3]x1    100.00/33.15
POL(c2(x1)) = x1    100.00/33.15
POL(c3(x1)) = x1    100.00/33.15
POL(c7(x1)) = x1    100.00/33.15
POL(mark(x1)) = [2] + x1    100.00/33.15
POL(ok(x1)) = [3] + x1   
100.00/33.15
100.00/33.15

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(f(a))) → mark(f(g(f(a)))) 100.00/33.15
active(f(z0)) → f(active(z0)) 100.00/33.15
f(mark(z0)) → mark(f(z0)) 100.00/33.15
f(ok(z0)) → ok(f(z0)) 100.00/33.15
proper(f(z0)) → f(proper(z0)) 100.00/33.15
proper(a) → ok(a) 100.00/33.15
proper(g(z0)) → g(proper(z0)) 100.00/33.15
g(ok(z0)) → ok(g(z0)) 100.00/33.15
top(mark(z0)) → top(proper(z0)) 100.00/33.15
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.15
F(ok(z0)) → c3(F(z0)) 100.00/33.15
G(ok(z0)) → c7(G(z0))
S tuples:none
K tuples:

F(mark(z0)) → c2(F(z0)) 100.00/33.15
F(ok(z0)) → c3(F(z0)) 100.00/33.15
G(ok(z0)) → c7(G(z0))
Defined Rule Symbols:

active, f, proper, g, top

Defined Pair Symbols:

F, G

Compound Symbols:

c2, c3, c7

100.00/33.15
100.00/33.15

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

The set S is empty
100.00/33.15
100.00/33.15

(50) BOUNDS(O(1), O(1))

100.00/33.15
100.00/33.15
100.25/33.21 EOF