YES(O(1), O(n^3)) 116.57/43.96 YES(O(1), O(n^3)) 116.57/43.97 116.57/43.97 116.57/43.97 116.57/43.97 116.57/43.97 116.57/43.97 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 116.57/43.97 116.57/43.97 116.57/43.97
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116.57/43.97
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^3)).

0 CpxTRS
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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
116.57/43.97 
↳2 CdtProblem
116.57/43.97 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
116.57/43.97 
↳4 CdtProblem
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↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
116.57/43.97 
↳6 CdtProblem
116.57/43.97 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
116.57/43.97 
↳8 CdtProblem
116.57/43.97 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
116.57/43.97 
↳10 CdtProblem
116.57/43.97 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
116.57/43.97 
↳12 CdtProblem
116.57/43.97 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
116.57/43.97 
↳14 CdtProblem
116.57/43.97 
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
116.57/43.97 
↳16 CdtProblem
116.57/43.97 
↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
116.57/43.97 
↳18 CdtProblem
116.57/43.97 
↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
116.57/43.97 
↳20 CdtProblem
116.57/43.97 
↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
116.57/43.97 
↳22 CdtProblem
116.57/43.97 
↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
116.57/43.97 
↳24 CdtProblem
116.57/43.97 
↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
116.57/43.97 
↳26 CdtProblem
116.57/43.97 
↳27 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
116.57/43.97 
↳28 CdtProblem
116.57/43.97 
↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
116.57/43.97 
↳30 CdtProblem
116.57/43.97 
↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
116.57/43.97 
↳32 CdtProblem
116.57/43.97 
↳33 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
116.57/43.97 
↳34 CdtProblem
116.57/43.97 
↳35 CdtUnreachableProof (⇔)
116.57/43.97 
↳36 CdtProblem
116.57/43.97 
↳37 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))))
116.57/43.97 
↳38 CdtProblem
116.57/43.97 
↳39 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳40 BOUNDS(O(1), O(1))
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116.57/43.97 116.57/43.97
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116.57/43.97

(0) Obligation:

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

active(f(X)) → mark(g(h(f(X)))) 116.57/43.97
active(f(X)) → f(active(X)) 116.57/43.97
active(h(X)) → h(active(X)) 116.57/43.97
f(mark(X)) → mark(f(X)) 116.57/43.97
h(mark(X)) → mark(h(X)) 116.57/43.97
proper(f(X)) → f(proper(X)) 116.57/43.97
proper(g(X)) → g(proper(X)) 116.57/43.97
proper(h(X)) → h(proper(X)) 116.57/43.97
f(ok(X)) → ok(f(X)) 116.57/43.97
g(ok(X)) → ok(g(X)) 116.57/43.97
h(ok(X)) → ok(h(X)) 116.57/43.97
top(mark(X)) → top(proper(X)) 116.57/43.97
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
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116.57/43.97

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

Converted CpxTRS to CDT
116.57/43.97
116.57/43.97

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.57/43.97
active(f(z0)) → f(active(z0)) 116.57/43.97
active(h(z0)) → h(active(z0)) 116.57/43.97
f(mark(z0)) → mark(f(z0)) 116.57/43.97
f(ok(z0)) → ok(f(z0)) 116.57/43.97
h(mark(z0)) → mark(h(z0)) 116.57/43.97
h(ok(z0)) → ok(h(z0)) 116.57/43.97
proper(f(z0)) → f(proper(z0)) 116.57/43.97
proper(g(z0)) → g(proper(z0)) 116.57/43.97
proper(h(z0)) → h(proper(z0)) 116.57/43.97
g(ok(z0)) → ok(g(z0)) 116.57/43.97
top(mark(z0)) → top(proper(z0)) 116.57/43.97
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0)) 116.57/43.97
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.57/43.97
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.57/43.97
F(mark(z0)) → c3(F(z0)) 116.57/43.97
F(ok(z0)) → c4(F(z0)) 116.57/43.97
H(mark(z0)) → c5(H(z0)) 116.57/43.97
H(ok(z0)) → c6(H(z0)) 116.57/43.97
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.57/43.97
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.57/43.97
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.57/43.97
G(ok(z0)) → c10(G(z0)) 116.57/43.97
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.57/43.97
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0)) 116.57/43.97
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.57/43.97
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.57/43.97
F(mark(z0)) → c3(F(z0)) 116.57/44.00
F(ok(z0)) → c4(F(z0)) 116.57/44.00
H(mark(z0)) → c5(H(z0)) 116.57/44.00
H(ok(z0)) → c6(H(z0)) 116.57/44.00
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.57/44.00
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.57/44.00
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.57/44.00
G(ok(z0)) → c10(G(z0)) 116.57/44.00
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.57/44.00
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

116.57/44.00
116.57/44.00

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

Removed 2 trailing tuple parts
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116.57/44.00

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.57/44.00
active(f(z0)) → f(active(z0)) 116.57/44.00
active(h(z0)) → h(active(z0)) 116.57/44.00
f(mark(z0)) → mark(f(z0)) 116.57/44.00
f(ok(z0)) → ok(f(z0)) 116.57/44.00
h(mark(z0)) → mark(h(z0)) 116.57/44.00
h(ok(z0)) → ok(h(z0)) 116.57/44.00
proper(f(z0)) → f(proper(z0)) 116.57/44.00
proper(g(z0)) → g(proper(z0)) 116.57/44.00
proper(h(z0)) → h(proper(z0)) 116.57/44.00
g(ok(z0)) → ok(g(z0)) 116.57/44.00
top(mark(z0)) → top(proper(z0)) 116.57/44.00
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.57/44.00
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.57/44.00
F(mark(z0)) → c3(F(z0)) 116.57/44.00
F(ok(z0)) → c4(F(z0)) 116.57/44.00
H(mark(z0)) → c5(H(z0)) 116.57/44.00
H(ok(z0)) → c6(H(z0)) 116.57/44.00
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.57/44.00
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.57/44.00
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.57/44.00
G(ok(z0)) → c10(G(z0)) 116.57/44.00
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.57/44.00
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.57/44.00
ACTIVE(f(z0)) → c(F(z0))
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.57/44.00
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.57/44.00
F(mark(z0)) → c3(F(z0)) 116.57/44.00
F(ok(z0)) → c4(F(z0)) 116.57/44.00
H(mark(z0)) → c5(H(z0)) 116.57/44.00
H(ok(z0)) → c6(H(z0)) 116.57/44.00
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.57/44.00
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.57/44.00
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.57/44.00
G(ok(z0)) → c10(G(z0)) 116.57/44.00
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.57/44.00
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.57/44.00
ACTIVE(f(z0)) → c(F(z0))
K tuples:none
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

116.57/44.00
116.57/44.00

(5) 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(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.57/44.00
ACTIVE(f(z0)) → c(F(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.57/44.00
active(f(z0)) → f(active(z0)) 116.57/44.00
active(h(z0)) → h(active(z0)) 116.57/44.00
h(mark(z0)) → mark(h(z0)) 116.57/44.00
h(ok(z0)) → ok(h(z0)) 116.57/44.00
f(mark(z0)) → mark(f(z0)) 116.57/44.00
f(ok(z0)) → ok(f(z0)) 116.57/44.00
proper(f(z0)) → f(proper(z0)) 116.57/44.00
proper(g(z0)) → g(proper(z0)) 116.57/44.00
proper(h(z0)) → h(proper(z0)) 116.57/44.00
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.57/44.00
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.57/44.00
F(mark(z0)) → c3(F(z0)) 116.57/44.00
F(ok(z0)) → c4(F(z0)) 116.57/44.00
H(mark(z0)) → c5(H(z0)) 116.57/44.00
H(ok(z0)) → c6(H(z0)) 116.57/44.00
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.57/44.00
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.57/44.00
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.57/44.00
G(ok(z0)) → c10(G(z0)) 116.57/44.00
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.57/44.00
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.57/44.00
ACTIVE(f(z0)) → c(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 116.57/44.00

POL(ACTIVE(x1)) = [4]    116.57/44.00
POL(F(x1)) = 0    116.57/44.00
POL(G(x1)) = 0    116.57/44.00
POL(H(x1)) = 0    116.57/44.00
POL(PROPER(x1)) = 0    116.57/44.00
POL(TOP(x1)) = [4]x1    116.57/44.00
POL(active(x1)) = 0    116.57/44.00
POL(c(x1)) = x1    116.57/44.00
POL(c1(x1, x2)) = x1 + x2    116.57/44.00
POL(c10(x1)) = x1    116.57/44.00
POL(c11(x1, x2)) = x1 + x2    116.57/44.00
POL(c12(x1, x2)) = x1 + x2    116.57/44.00
POL(c2(x1, x2)) = x1 + x2    116.57/44.00
POL(c3(x1)) = x1    116.57/44.00
POL(c4(x1)) = x1    116.57/44.00
POL(c5(x1)) = x1    116.57/44.00
POL(c6(x1)) = x1    116.57/44.00
POL(c7(x1, x2)) = x1 + x2    116.57/44.00
POL(c8(x1, x2)) = x1 + x2    116.57/44.00
POL(c9(x1, x2)) = x1 + x2    116.57/44.00
POL(f(x1)) = [2]x1    116.57/44.00
POL(g(x1)) = [4]x1    116.57/44.00
POL(h(x1)) = [2]x1    116.57/44.00
POL(mark(x1)) = 0    116.57/44.00
POL(ok(x1)) = [4]    116.57/44.00
POL(proper(x1)) = 0   
116.86/44.01
116.86/44.01

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.01
active(f(z0)) → f(active(z0)) 116.86/44.01
active(h(z0)) → h(active(z0)) 116.86/44.01
f(mark(z0)) → mark(f(z0)) 116.86/44.01
f(ok(z0)) → ok(f(z0)) 116.86/44.01
h(mark(z0)) → mark(h(z0)) 116.86/44.01
h(ok(z0)) → ok(h(z0)) 116.86/44.01
proper(f(z0)) → f(proper(z0)) 116.86/44.01
proper(g(z0)) → g(proper(z0)) 116.86/44.01
proper(h(z0)) → h(proper(z0)) 116.86/44.01
g(ok(z0)) → ok(g(z0)) 116.86/44.01
top(mark(z0)) → top(proper(z0)) 116.86/44.01
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.01
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.01
F(mark(z0)) → c3(F(z0)) 116.86/44.01
F(ok(z0)) → c4(F(z0)) 116.86/44.01
H(mark(z0)) → c5(H(z0)) 116.86/44.01
H(ok(z0)) → c6(H(z0)) 116.86/44.01
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.01
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.01
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.01
G(ok(z0)) → c10(G(z0)) 116.86/44.01
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.01
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.01
ACTIVE(f(z0)) → c(F(z0))
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.01
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.01
F(mark(z0)) → c3(F(z0)) 116.86/44.01
F(ok(z0)) → c4(F(z0)) 116.86/44.01
H(mark(z0)) → c5(H(z0)) 116.86/44.01
H(ok(z0)) → c6(H(z0)) 116.86/44.01
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

116.86/44.02
116.86/44.02

(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.

TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 116.86/44.02

POL(ACTIVE(x1)) = 0    116.86/44.02
POL(F(x1)) = 0    116.86/44.02
POL(G(x1)) = 0    116.86/44.02
POL(H(x1)) = 0    116.86/44.02
POL(PROPER(x1)) = [1]    116.86/44.02
POL(TOP(x1)) = [4]x1    116.86/44.02
POL(active(x1)) = x1    116.86/44.02
POL(c(x1)) = x1    116.86/44.02
POL(c1(x1, x2)) = x1 + x2    116.86/44.02
POL(c10(x1)) = x1    116.86/44.02
POL(c11(x1, x2)) = x1 + x2    116.86/44.02
POL(c12(x1, x2)) = x1 + x2    116.86/44.02
POL(c2(x1, x2)) = x1 + x2    116.86/44.02
POL(c3(x1)) = x1    116.86/44.02
POL(c4(x1)) = x1    116.86/44.02
POL(c5(x1)) = x1    116.86/44.02
POL(c6(x1)) = x1    116.86/44.02
POL(c7(x1, x2)) = x1 + x2    116.86/44.02
POL(c8(x1, x2)) = x1 + x2    116.86/44.02
POL(c9(x1, x2)) = x1 + x2    116.86/44.02
POL(f(x1)) = [2] + x1    116.86/44.02
POL(g(x1)) = 0    116.86/44.02
POL(h(x1)) = [2]x1    116.86/44.02
POL(mark(x1)) = [2] + x1    116.86/44.02
POL(ok(x1)) = x1    116.86/44.02
POL(proper(x1)) = x1   
116.86/44.02
116.86/44.02

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

116.86/44.02
116.86/44.02

(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.

G(ok(z0)) → c10(G(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 116.86/44.02

POL(ACTIVE(x1)) = 0    116.86/44.02
POL(F(x1)) = 0    116.86/44.02
POL(G(x1)) = x1    116.86/44.02
POL(H(x1)) = 0    116.86/44.02
POL(PROPER(x1)) = 0    116.86/44.02
POL(TOP(x1)) = 0    116.86/44.02
POL(active(x1)) = 0    116.86/44.02
POL(c(x1)) = x1    116.86/44.02
POL(c1(x1, x2)) = x1 + x2    116.86/44.02
POL(c10(x1)) = x1    116.86/44.02
POL(c11(x1, x2)) = x1 + x2    116.86/44.02
POL(c12(x1, x2)) = x1 + x2    116.86/44.02
POL(c2(x1, x2)) = x1 + x2    116.86/44.02
POL(c3(x1)) = x1    116.86/44.02
POL(c4(x1)) = x1    116.86/44.02
POL(c5(x1)) = x1    116.86/44.02
POL(c6(x1)) = x1    116.86/44.02
POL(c7(x1, x2)) = x1 + x2    116.86/44.02
POL(c8(x1, x2)) = x1 + x2    116.86/44.02
POL(c9(x1, x2)) = x1 + x2    116.86/44.02
POL(f(x1)) = [4]x1    116.86/44.02
POL(g(x1)) = [4]x1    116.86/44.02
POL(h(x1)) = x1    116.86/44.02
POL(mark(x1)) = [4]    116.86/44.02
POL(ok(x1)) = [1] + x1    116.86/44.02
POL(proper(x1)) = 0   
116.86/44.02
116.86/44.02

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

116.86/44.02
116.86/44.02

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 116.86/44.02

POL(ACTIVE(x1)) = 0    116.86/44.02
POL(F(x1)) = 0    116.86/44.02
POL(G(x1)) = 0    116.86/44.02
POL(H(x1)) = 0    116.86/44.02
POL(PROPER(x1)) = [2]x1    116.86/44.02
POL(TOP(x1)) = [2]x12    116.86/44.02
POL(active(x1)) = [2] + x1    116.86/44.02
POL(c(x1)) = x1    116.86/44.02
POL(c1(x1, x2)) = x1 + x2    116.86/44.02
POL(c10(x1)) = x1    116.86/44.02
POL(c11(x1, x2)) = x1 + x2    116.86/44.02
POL(c12(x1, x2)) = x1 + x2    116.86/44.02
POL(c2(x1, x2)) = x1 + x2    116.86/44.02
POL(c3(x1)) = x1    116.86/44.02
POL(c4(x1)) = x1    116.86/44.02
POL(c5(x1)) = x1    116.86/44.02
POL(c6(x1)) = x1    116.86/44.02
POL(c7(x1, x2)) = x1 + x2    116.86/44.02
POL(c8(x1, x2)) = x1 + x2    116.86/44.02
POL(c9(x1, x2)) = x1 + x2    116.86/44.02
POL(f(x1)) = [1] + x1    116.86/44.02
POL(g(x1)) = [1] + x1    116.86/44.02
POL(h(x1)) = x1    116.86/44.02
POL(mark(x1)) = [1] + x1    116.86/44.02
POL(ok(x1)) = [2] + x1    116.86/44.02
POL(proper(x1)) = x1   
116.86/44.02
116.86/44.02

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

116.86/44.02
116.86/44.02

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 116.86/44.02

POL(ACTIVE(x1)) = [3] + x1    116.86/44.02
POL(F(x1)) = 0    116.86/44.02
POL(G(x1)) = 0    116.86/44.02
POL(H(x1)) = 0    116.86/44.02
POL(PROPER(x1)) = 0    116.86/44.02
POL(TOP(x1)) = x12    116.86/44.02
POL(active(x1)) = [1] + x1    116.86/44.02
POL(c(x1)) = x1    116.86/44.02
POL(c1(x1, x2)) = x1 + x2    116.86/44.02
POL(c10(x1)) = x1    116.86/44.02
POL(c11(x1, x2)) = x1 + x2    116.86/44.02
POL(c12(x1, x2)) = x1 + x2    116.86/44.02
POL(c2(x1, x2)) = x1 + x2    116.86/44.02
POL(c3(x1)) = x1    116.86/44.02
POL(c4(x1)) = x1    116.86/44.02
POL(c5(x1)) = x1    116.86/44.02
POL(c6(x1)) = x1    116.86/44.02
POL(c7(x1, x2)) = x1 + x2    116.86/44.02
POL(c8(x1, x2)) = x1 + x2    116.86/44.02
POL(c9(x1, x2)) = x1 + x2    116.86/44.02
POL(f(x1)) = x1    116.86/44.02
POL(g(x1)) = x1    116.86/44.02
POL(h(x1)) = [1] + x1    116.86/44.02
POL(mark(x1)) = x1    116.86/44.02
POL(ok(x1)) = [2] + x1    116.86/44.02
POL(proper(x1)) = x1   
116.86/44.02
116.86/44.02

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

116.86/44.02
116.86/44.02

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 116.86/44.02

POL(ACTIVE(x1)) = [1] + x1    116.86/44.02
POL(F(x1)) = 0    116.86/44.02
POL(G(x1)) = 0    116.86/44.02
POL(H(x1)) = 0    116.86/44.02
POL(PROPER(x1)) = 0    116.86/44.02
POL(TOP(x1)) = x12    116.86/44.02
POL(active(x1)) = x1    116.86/44.02
POL(c(x1)) = x1    116.86/44.02
POL(c1(x1, x2)) = x1 + x2    116.86/44.02
POL(c10(x1)) = x1    116.86/44.02
POL(c11(x1, x2)) = x1 + x2    116.86/44.02
POL(c12(x1, x2)) = x1 + x2    116.86/44.02
POL(c2(x1, x2)) = x1 + x2    116.86/44.02
POL(c3(x1)) = x1    116.86/44.02
POL(c4(x1)) = x1    116.86/44.02
POL(c5(x1)) = x1    116.86/44.02
POL(c6(x1)) = x1    116.86/44.02
POL(c7(x1, x2)) = x1 + x2    116.86/44.02
POL(c8(x1, x2)) = x1 + x2    116.86/44.02
POL(c9(x1, x2)) = x1 + x2    116.86/44.02
POL(f(x1)) = [2] + x1    116.86/44.02
POL(g(x1)) = x1    116.86/44.02
POL(h(x1)) = x1    116.86/44.02
POL(mark(x1)) = x1    116.86/44.02
POL(ok(x1)) = [1] + x1    116.86/44.02
POL(proper(x1)) = x1   
116.86/44.02
116.86/44.02

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
S tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

116.86/44.02
116.86/44.02

(17) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

F(ok(z0)) → c4(F(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 116.86/44.02

POL(ACTIVE(x1)) = [1] + x1    116.86/44.02
POL(F(x1)) = [2]x12    116.86/44.02
POL(G(x1)) = 0    116.86/44.02
POL(H(x1)) = 0    116.86/44.02
POL(PROPER(x1)) = 0    116.86/44.02
POL(TOP(x1)) = x12    116.86/44.02
POL(active(x1)) = x1    116.86/44.02
POL(c(x1)) = x1    116.86/44.02
POL(c1(x1, x2)) = x1 + x2    116.86/44.02
POL(c10(x1)) = x1    116.86/44.02
POL(c11(x1, x2)) = x1 + x2    116.86/44.02
POL(c12(x1, x2)) = x1 + x2    116.86/44.02
POL(c2(x1, x2)) = x1 + x2    116.86/44.02
POL(c3(x1)) = x1    116.86/44.02
POL(c4(x1)) = x1    116.86/44.02
POL(c5(x1)) = x1    116.86/44.02
POL(c6(x1)) = x1    116.86/44.02
POL(c7(x1, x2)) = x1 + x2    116.86/44.02
POL(c8(x1, x2)) = x1 + x2    116.86/44.02
POL(c9(x1, x2)) = x1 + x2    116.86/44.02
POL(f(x1)) = x1 + [2]x12    116.86/44.02
POL(g(x1)) = x1    116.86/44.02
POL(h(x1)) = x1    116.86/44.02
POL(mark(x1)) = x1    116.86/44.02
POL(ok(x1)) = [1] + x1    116.86/44.02
POL(proper(x1)) = 0   
116.86/44.02
116.86/44.02

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
S tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

116.86/44.02
116.86/44.02

(19) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0))
And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 116.86/44.02

POL(ACTIVE(x1)) = 0    116.86/44.02
POL(F(x1)) = 0    116.86/44.02
POL(G(x1)) = 0    116.86/44.02
POL(H(x1)) = 0    116.86/44.02
POL(PROPER(x1)) = [2]x1    116.86/44.02
POL(TOP(x1)) = [2]x12    116.86/44.02
POL(active(x1)) = [3] + x1    116.86/44.02
POL(c(x1)) = x1    116.86/44.02
POL(c1(x1, x2)) = x1 + x2    116.86/44.02
POL(c10(x1)) = x1    116.86/44.02
POL(c11(x1, x2)) = x1 + x2    116.86/44.02
POL(c12(x1, x2)) = x1 + x2    116.86/44.02
POL(c2(x1, x2)) = x1 + x2    116.86/44.02
POL(c3(x1)) = x1    116.86/44.02
POL(c4(x1)) = x1    116.86/44.02
POL(c5(x1)) = x1    116.86/44.02
POL(c6(x1)) = x1    116.86/44.02
POL(c7(x1, x2)) = x1 + x2    116.86/44.02
POL(c8(x1, x2)) = x1 + x2    116.86/44.02
POL(c9(x1, x2)) = x1 + x2    116.86/44.02
POL(f(x1)) = x1    116.86/44.02
POL(g(x1)) = x1    116.86/44.02
POL(h(x1)) = [2] + x1    116.86/44.02
POL(mark(x1)) = [1] + x1    116.86/44.02
POL(ok(x1)) = [3] + x1    116.86/44.02
POL(proper(x1)) = x1   
116.86/44.02
116.86/44.02

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0))
S tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

116.86/44.02
116.86/44.02

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

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

ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
116.86/44.02
116.86/44.02

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
S tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c, c1

116.86/44.02
116.86/44.02

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

Use narrowing to replace ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) by

ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
116.86/44.02
116.86/44.02

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
S tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c, c1, c2

116.86/44.02
116.86/44.02

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

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

PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
116.86/44.02
116.86/44.02

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0))) 116.86/44.02
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
S tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c8, c9, c10, c11, c12, c, c1, c2, c7

116.86/44.02
116.86/44.02

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

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

PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))
116.86/44.02
116.86/44.02

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0))) 116.86/44.02
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))
S tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c9, c10, c11, c12, c, c1, c2, c7, c8

116.86/44.02
116.86/44.02

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

Use narrowing to replace PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) by

PROPER(h(f(z0))) → c9(H(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(h(g(z0))) → c9(H(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(h(h(z0))) → c9(H(h(proper(z0))), PROPER(h(z0)))
116.86/44.02
116.86/44.02

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0))) 116.86/44.02
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
PROPER(h(f(z0))) → c9(H(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(h(g(z0))) → c9(H(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(h(h(z0))) → c9(H(h(proper(z0))), PROPER(h(z0)))
S tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, G, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c5, c6, c10, c11, c12, c, c1, c2, c7, c8, c9

116.86/44.02
116.86/44.02

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

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

TOP(mark(f(z0))) → c11(TOP(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
TOP(mark(g(z0))) → c11(TOP(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
TOP(mark(h(z0))) → c11(TOP(h(proper(z0))), PROPER(h(z0)))
116.86/44.02
116.86/44.02

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0))) 116.86/44.02
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
PROPER(h(f(z0))) → c9(H(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(h(g(z0))) → c9(H(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(h(h(z0))) → c9(H(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
TOP(mark(f(z0))) → c11(TOP(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
TOP(mark(g(z0))) → c11(TOP(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
TOP(mark(h(z0))) → c11(TOP(h(proper(z0))), PROPER(h(z0)))
S tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, G, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c5, c6, c10, c12, c, c1, c2, c7, c8, c9, c11

116.86/44.02
116.86/44.02

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

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

TOP(ok(f(z0))) → c12(TOP(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
TOP(ok(f(z0))) → c12(TOP(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
TOP(ok(h(z0))) → c12(TOP(h(active(z0))), ACTIVE(h(z0)))
116.86/44.02
116.86/44.02

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0))) 116.86/44.02
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
PROPER(h(f(z0))) → c9(H(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(h(g(z0))) → c9(H(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(h(h(z0))) → c9(H(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
TOP(mark(f(z0))) → c11(TOP(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
TOP(mark(g(z0))) → c11(TOP(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
TOP(mark(h(z0))) → c11(TOP(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
TOP(ok(f(z0))) → c12(TOP(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
TOP(ok(f(z0))) → c12(TOP(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
TOP(ok(h(z0))) → c12(TOP(h(active(z0))), ACTIVE(h(z0)))
S tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0))
K tuples:

TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0)) 116.86/44.02
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) 116.86/44.02
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) 116.86/44.02
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) 116.86/44.02
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, G, ACTIVE, PROPER, TOP

Compound Symbols:

c3, c4, c5, c6, c10, c, c1, c2, c7, c8, c9, c11, c12

116.86/44.02
116.86/44.02

(35) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(f(z0)) → c(F(z0)) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0))) 116.86/44.02
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
PROPER(h(f(z0))) → c9(H(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
PROPER(h(g(z0))) → c9(H(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
PROPER(h(h(z0))) → c9(H(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
TOP(mark(f(z0))) → c11(TOP(f(proper(z0))), PROPER(f(z0))) 116.86/44.02
TOP(mark(g(z0))) → c11(TOP(g(proper(z0))), PROPER(g(z0))) 116.86/44.02
TOP(mark(h(z0))) → c11(TOP(h(proper(z0))), PROPER(h(z0))) 116.86/44.02
TOP(ok(f(z0))) → c12(TOP(mark(g(h(f(z0))))), ACTIVE(f(z0))) 116.86/44.02
TOP(ok(f(z0))) → c12(TOP(f(active(z0))), ACTIVE(f(z0))) 116.86/44.02
TOP(ok(h(z0))) → c12(TOP(h(active(z0))), ACTIVE(h(z0)))
116.86/44.02
116.86/44.02

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0))
S tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0))
K tuples:

G(ok(z0)) → c10(G(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, G

Compound Symbols:

c3, c4, c5, c6, c10

116.86/44.02
116.86/44.02

(37) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

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

F(mark(z0)) → c3(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 116.86/44.02

POL(F(x1)) = x1 + x12 + x13    116.86/44.02
POL(G(x1)) = x1 + x13    116.86/44.02
POL(H(x1)) = x1 + x12 + x13    116.86/44.02
POL(c10(x1)) = x1    116.86/44.02
POL(c3(x1)) = x1    116.86/44.02
POL(c4(x1)) = x1    116.86/44.02
POL(c5(x1)) = x1    116.86/44.02
POL(c6(x1)) = x1    116.86/44.02
POL(mark(x1)) = [1] + x1    116.86/44.02
POL(ok(x1)) = [1] + x1   
116.86/44.02
116.86/44.02

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0)))) 116.86/44.02
active(f(z0)) → f(active(z0)) 116.86/44.02
active(h(z0)) → h(active(z0)) 116.86/44.02
f(mark(z0)) → mark(f(z0)) 116.86/44.02
f(ok(z0)) → ok(f(z0)) 116.86/44.02
h(mark(z0)) → mark(h(z0)) 116.86/44.02
h(ok(z0)) → ok(h(z0)) 116.86/44.02
proper(f(z0)) → f(proper(z0)) 116.86/44.02
proper(g(z0)) → g(proper(z0)) 116.86/44.02
proper(h(z0)) → h(proper(z0)) 116.86/44.02
g(ok(z0)) → ok(g(z0)) 116.86/44.02
top(mark(z0)) → top(proper(z0)) 116.86/44.02
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c3(F(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0)) 116.86/44.02
G(ok(z0)) → c10(G(z0))
S tuples:none
K tuples:

G(ok(z0)) → c10(G(z0)) 116.86/44.02
F(ok(z0)) → c4(F(z0)) 116.86/44.02
F(mark(z0)) → c3(F(z0)) 116.86/44.02
H(mark(z0)) → c5(H(z0)) 116.86/44.02
H(ok(z0)) → c6(H(z0))
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, G

Compound Symbols:

c3, c4, c5, c6, c10

116.86/44.02
116.86/44.02

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

The set S is empty
116.86/44.02
116.86/44.02

(40) BOUNDS(O(1), O(1))

116.86/44.02
116.86/44.02
116.86/44.09 EOF