YES(O(1), O(n^2)) 152.92/51.67 YES(O(1), O(n^2)) 153.29/51.72 153.29/51.72 153.29/51.72 153.29/51.72 153.29/51.72 153.29/51.72 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 153.29/51.72 153.29/51.72 153.29/51.72
153.29/51.72
153.29/51.72
153.29/51.72
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
153.29/51.72 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳2 CdtProblem
153.29/51.72 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳4 CdtProblem
153.29/51.72 
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳6 CdtProblem
153.29/51.72 
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳8 CdtProblem
153.29/51.72 
↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳10 CdtProblem
153.29/51.72 
↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳12 CdtProblem
153.29/51.72 
↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳14 CdtProblem
153.29/51.72 
↳15 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳16 CdtProblem
153.29/51.72 
↳17 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳18 CdtProblem
153.29/51.72 
↳19 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳20 CdtProblem
153.29/51.72 
↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳22 CdtProblem
153.29/51.72 
↳23 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳24 CdtProblem
153.29/51.72 
↳25 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳26 CdtProblem
153.29/51.72 
↳27 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳28 CdtProblem
153.29/51.72 
↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳30 CdtProblem
153.29/51.72 
↳31 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳32 CdtProblem
153.29/51.72 
↳33 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳34 CdtProblem
153.29/51.72 
↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳36 CdtProblem
153.29/51.72 
↳37 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳38 CdtProblem
153.29/51.72 
↳39 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳40 CdtProblem
153.29/51.72 
↳41 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳42 CdtProblem
153.29/51.72 
↳43 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳44 CdtProblem
153.29/51.72 
↳45 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳46 CdtProblem
153.29/51.72 
↳47 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳48 CdtProblem
153.29/51.72 
↳49 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳50 CdtProblem
153.29/51.72 
↳51 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳52 CdtProblem
153.29/51.72 
↳53 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
153.29/51.72 
↳54 CdtProblem
153.29/51.72 
↳55 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
153.29/51.72 
↳56 CdtProblem
153.29/51.72 
↳57 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳58 CdtProblem
153.29/51.72 
↳59 CdtUnreachableProof (⇔)
153.29/51.72 
↳60 CdtProblem
153.29/51.72 
↳61 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳62 CdtProblem
153.29/51.72 
↳63 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳64 CdtProblem
153.29/51.72 
↳65 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
153.29/51.72 
↳66 CdtProblem
153.29/51.72 
↳67 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
153.29/51.72 
↳68 CdtProblem
153.29/51.72 
↳69 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
153.29/51.72 
↳70 CdtProblem
153.29/51.72 
↳71 SIsEmptyProof (BOTH BOUNDS(ID, ID))
153.29/51.72 
↳72 BOUNDS(O(1), O(1))
153.29/51.72
153.29/51.72 153.29/51.72
153.29/51.72
153.29/51.72

(0) Obligation:

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

active(h(X)) → mark(g(X, X)) 153.29/51.72
active(g(a, X)) → mark(f(b, X)) 153.29/51.72
active(f(X, X)) → mark(h(a)) 153.29/51.72
active(a) → mark(b) 153.29/51.72
active(h(X)) → h(active(X)) 153.29/51.72
active(g(X1, X2)) → g(active(X1), X2) 153.29/51.72
active(f(X1, X2)) → f(active(X1), X2) 153.29/51.72
h(mark(X)) → mark(h(X)) 153.29/51.72
g(mark(X1), X2) → mark(g(X1, X2)) 153.29/51.72
f(mark(X1), X2) → mark(f(X1, X2)) 153.29/51.72
proper(h(X)) → h(proper(X)) 153.29/51.72
proper(g(X1, X2)) → g(proper(X1), proper(X2)) 153.29/51.72
proper(a) → ok(a) 153.29/51.72
proper(f(X1, X2)) → f(proper(X1), proper(X2)) 153.29/51.72
proper(b) → ok(b) 153.29/51.72
h(ok(X)) → ok(h(X)) 153.29/51.72
g(ok(X1), ok(X2)) → ok(g(X1, X2)) 153.29/51.72
f(ok(X1), ok(X2)) → ok(f(X1, X2)) 153.29/51.72
top(mark(X)) → top(proper(X)) 153.29/51.72
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
153.29/51.72
153.29/51.72

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

Converted CpxTRS to CDT
153.29/51.72
153.29/51.72

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 153.29/51.72
active(g(a, z0)) → mark(f(b, z0)) 153.29/51.72
active(f(z0, z0)) → mark(h(a)) 153.29/51.72
active(a) → mark(b) 153.29/51.72
active(h(z0)) → h(active(z0)) 153.29/51.72
active(g(z0, z1)) → g(active(z0), z1) 153.29/51.72
active(f(z0, z1)) → f(active(z0), z1) 153.29/51.72
h(mark(z0)) → mark(h(z0)) 153.29/51.72
h(ok(z0)) → ok(h(z0)) 153.29/51.72
g(mark(z0), z1) → mark(g(z0, z1)) 153.29/51.72
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 153.29/51.72
f(mark(z0), z1) → mark(f(z0, z1)) 153.29/51.75
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 153.29/51.75
proper(h(z0)) → h(proper(z0)) 153.29/51.75
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 153.29/51.75
proper(a) → ok(a) 153.29/51.75
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 153.29/51.75
proper(b) → ok(b) 153.29/51.75
top(mark(z0)) → top(proper(z0)) 153.29/51.75
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.75
ACTIVE(g(a, z0)) → c1(F(b, z0)) 153.29/51.75
ACTIVE(f(z0, z0)) → c2(H(a)) 153.29/51.75
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) 153.29/51.75
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.75
H(mark(z0)) → c7(H(z0)) 153.29/51.75
H(ok(z0)) → c8(H(z0)) 153.29/51.75
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.75
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.75
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.75
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.75
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.75
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.75
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.75
ACTIVE(g(a, z0)) → c1(F(b, z0)) 153.29/51.75
ACTIVE(f(z0, z0)) → c2(H(a)) 153.29/51.75
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) 153.29/51.75
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.75
H(mark(z0)) → c7(H(z0)) 153.29/51.75
H(ok(z0)) → c8(H(z0)) 153.29/51.75
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.75
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.75
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.75
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.75
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.75
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.75
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c1, c2, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19

153.29/51.75
153.29/51.75

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

Removed 2 trailing tuple parts
153.29/51.75
153.29/51.75

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 153.29/51.75
active(g(a, z0)) → mark(f(b, z0)) 153.29/51.75
active(f(z0, z0)) → mark(h(a)) 153.29/51.75
active(a) → mark(b) 153.29/51.75
active(h(z0)) → h(active(z0)) 153.29/51.75
active(g(z0, z1)) → g(active(z0), z1) 153.29/51.75
active(f(z0, z1)) → f(active(z0), z1) 153.29/51.75
h(mark(z0)) → mark(h(z0)) 153.29/51.75
h(ok(z0)) → ok(h(z0)) 153.29/51.75
g(mark(z0), z1) → mark(g(z0, z1)) 153.29/51.75
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 153.29/51.75
f(mark(z0), z1) → mark(f(z0, z1)) 153.29/51.75
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 153.29/51.75
proper(h(z0)) → h(proper(z0)) 153.29/51.75
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 153.29/51.75
proper(a) → ok(a) 153.29/51.75
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 153.29/51.75
proper(b) → ok(b) 153.29/51.75
top(mark(z0)) → top(proper(z0)) 153.29/51.75
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.75
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) 153.29/51.75
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.75
H(mark(z0)) → c7(H(z0)) 153.29/51.75
H(ok(z0)) → c8(H(z0)) 153.29/51.75
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.75
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.75
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.75
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.75
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.75
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.75
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(a, z0)) → c1 153.29/51.75
ACTIVE(f(z0, z0)) → c2
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.75
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) 153.29/51.75
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.75
H(mark(z0)) → c7(H(z0)) 153.29/51.75
H(ok(z0)) → c8(H(z0)) 153.29/51.75
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.75
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.75
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.75
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.75
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.75
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.75
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(a, z0)) → c1 153.29/51.75
ACTIVE(f(z0, z0)) → c2
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c1, c2

153.29/51.75
153.29/51.75

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

Removed 2 trailing nodes:

ACTIVE(g(a, z0)) → c1 153.29/51.75
ACTIVE(f(z0, z0)) → c2
153.29/51.75
153.29/51.75

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 153.29/51.75
active(g(a, z0)) → mark(f(b, z0)) 153.29/51.75
active(f(z0, z0)) → mark(h(a)) 153.29/51.75
active(a) → mark(b) 153.29/51.75
active(h(z0)) → h(active(z0)) 153.29/51.75
active(g(z0, z1)) → g(active(z0), z1) 153.29/51.75
active(f(z0, z1)) → f(active(z0), z1) 153.29/51.75
h(mark(z0)) → mark(h(z0)) 153.29/51.75
h(ok(z0)) → ok(h(z0)) 153.29/51.75
g(mark(z0), z1) → mark(g(z0, z1)) 153.29/51.75
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 153.29/51.75
f(mark(z0), z1) → mark(f(z0, z1)) 153.29/51.75
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 153.29/51.75
proper(h(z0)) → h(proper(z0)) 153.29/51.75
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 153.29/51.75
proper(a) → ok(a) 153.29/51.75
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 153.29/51.75
proper(b) → ok(b) 153.29/51.75
top(mark(z0)) → top(proper(z0)) 153.29/51.75
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.75
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) 153.29/51.75
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.75
H(mark(z0)) → c7(H(z0)) 153.29/51.75
H(ok(z0)) → c8(H(z0)) 153.29/51.75
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.75
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.75
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.75
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.75
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.75
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.75
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(a, z0)) → c1 153.29/51.75
ACTIVE(f(z0, z0)) → c2
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.75
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) 153.29/51.75
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.75
H(mark(z0)) → c7(H(z0)) 153.29/51.75
H(ok(z0)) → c8(H(z0)) 153.29/51.75
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.75
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.75
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.75
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.75
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.75
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.75
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(a, z0)) → c1 153.29/51.75
ACTIVE(f(z0, z0)) → c2
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c1, c2

153.29/51.75
153.29/51.75

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

Use narrowing to replace ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0)) by

ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0))) 153.29/51.75
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.75
ACTIVE(h(a)) → c4(H(mark(b)), ACTIVE(a)) 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.75
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
153.29/51.75
153.29/51.75

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 153.29/51.75
active(g(a, z0)) → mark(f(b, z0)) 153.29/51.75
active(f(z0, z0)) → mark(h(a)) 153.29/51.75
active(a) → mark(b) 153.29/51.75
active(h(z0)) → h(active(z0)) 153.29/51.75
active(g(z0, z1)) → g(active(z0), z1) 153.29/51.75
active(f(z0, z1)) → f(active(z0), z1) 153.29/51.75
h(mark(z0)) → mark(h(z0)) 153.29/51.75
h(ok(z0)) → ok(h(z0)) 153.29/51.75
g(mark(z0), z1) → mark(g(z0, z1)) 153.29/51.75
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 153.29/51.75
f(mark(z0), z1) → mark(f(z0, z1)) 153.29/51.75
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 153.29/51.75
proper(h(z0)) → h(proper(z0)) 153.29/51.75
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 153.29/51.75
proper(a) → ok(a) 153.29/51.75
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 153.29/51.75
proper(b) → ok(b) 153.29/51.75
top(mark(z0)) → top(proper(z0)) 153.29/51.75
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.75
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) 153.29/51.75
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.75
H(mark(z0)) → c7(H(z0)) 153.29/51.75
H(ok(z0)) → c8(H(z0)) 153.29/51.75
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.75
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.75
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.75
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.75
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.75
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.75
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(a, z0)) → c1 153.29/51.75
ACTIVE(f(z0, z0)) → c2 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0))) 153.29/51.75
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.75
ACTIVE(h(a)) → c4(H(mark(b)), ACTIVE(a)) 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.75
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.75
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) 153.29/51.75
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.75
H(mark(z0)) → c7(H(z0)) 153.29/51.75
H(ok(z0)) → c8(H(z0)) 153.29/51.75
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.75
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.75
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.75
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.75
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.75
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.75
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(a, z0)) → c1 153.29/51.75
ACTIVE(f(z0, z0)) → c2 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0))) 153.29/51.75
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.75
ACTIVE(h(a)) → c4(H(mark(b)), ACTIVE(a)) 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.75
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c1, c2, c4

153.29/51.75
153.29/51.75

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

Removed 1 trailing tuple parts
153.29/51.75
153.29/51.75

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 153.29/51.75
active(g(a, z0)) → mark(f(b, z0)) 153.29/51.75
active(f(z0, z0)) → mark(h(a)) 153.29/51.75
active(a) → mark(b) 153.29/51.75
active(h(z0)) → h(active(z0)) 153.29/51.75
active(g(z0, z1)) → g(active(z0), z1) 153.29/51.75
active(f(z0, z1)) → f(active(z0), z1) 153.29/51.75
h(mark(z0)) → mark(h(z0)) 153.29/51.75
h(ok(z0)) → ok(h(z0)) 153.29/51.75
g(mark(z0), z1) → mark(g(z0, z1)) 153.29/51.75
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 153.29/51.75
f(mark(z0), z1) → mark(f(z0, z1)) 153.29/51.75
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 153.29/51.75
proper(h(z0)) → h(proper(z0)) 153.29/51.75
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 153.29/51.75
proper(a) → ok(a) 153.29/51.75
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 153.29/51.75
proper(b) → ok(b) 153.29/51.75
top(mark(z0)) → top(proper(z0)) 153.29/51.75
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.75
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) 153.29/51.75
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.75
H(mark(z0)) → c7(H(z0)) 153.29/51.75
H(ok(z0)) → c8(H(z0)) 153.29/51.75
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.75
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.75
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.75
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.75
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.75
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.75
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(a, z0)) → c1 153.29/51.75
ACTIVE(f(z0, z0)) → c2 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0))) 153.29/51.75
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.75
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.29/51.75
ACTIVE(h(a)) → c4(H(mark(b)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.75
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) 153.29/51.75
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.75
H(mark(z0)) → c7(H(z0)) 153.29/51.75
H(ok(z0)) → c8(H(z0)) 153.29/51.75
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.75
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.75
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.75
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.75
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.75
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.75
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(a, z0)) → c1 153.29/51.75
ACTIVE(f(z0, z0)) → c2 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0))) 153.29/51.75
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.75
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.29/51.75
ACTIVE(h(a)) → c4(H(mark(b)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c1, c2, c4, c4

153.29/51.75
153.29/51.75

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

Removed 2 trailing nodes:

ACTIVE(g(a, z0)) → c1 153.29/51.75
ACTIVE(f(z0, z0)) → c2
153.29/51.75
153.29/51.75

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 153.29/51.75
active(g(a, z0)) → mark(f(b, z0)) 153.29/51.75
active(f(z0, z0)) → mark(h(a)) 153.29/51.75
active(a) → mark(b) 153.29/51.75
active(h(z0)) → h(active(z0)) 153.29/51.75
active(g(z0, z1)) → g(active(z0), z1) 153.29/51.75
active(f(z0, z1)) → f(active(z0), z1) 153.29/51.75
h(mark(z0)) → mark(h(z0)) 153.29/51.75
h(ok(z0)) → ok(h(z0)) 153.29/51.75
g(mark(z0), z1) → mark(g(z0, z1)) 153.29/51.75
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 153.29/51.75
f(mark(z0), z1) → mark(f(z0, z1)) 153.29/51.75
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 153.29/51.75
proper(h(z0)) → h(proper(z0)) 153.29/51.75
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 153.29/51.75
proper(a) → ok(a) 153.29/51.75
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 153.29/51.75
proper(b) → ok(b) 153.29/51.75
top(mark(z0)) → top(proper(z0)) 153.29/51.75
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.75
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) 153.29/51.75
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.75
H(mark(z0)) → c7(H(z0)) 153.29/51.75
H(ok(z0)) → c8(H(z0)) 153.29/51.75
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.75
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.75
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.75
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.75
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.75
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.75
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(a, z0)) → c1 153.29/51.75
ACTIVE(f(z0, z0)) → c2 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0))) 153.29/51.75
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.75
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.29/51.75
ACTIVE(h(a)) → c4(H(mark(b)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.75
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) 153.29/51.75
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.75
H(mark(z0)) → c7(H(z0)) 153.29/51.75
H(ok(z0)) → c8(H(z0)) 153.29/51.75
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.75
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.75
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.75
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.75
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.75
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.75
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(a, z0)) → c1 153.29/51.75
ACTIVE(f(z0, z0)) → c2 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0))) 153.29/51.75
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.75
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.29/51.75
ACTIVE(h(a)) → c4(H(mark(b)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c1, c2, c4, c4

153.29/51.75
153.29/51.75

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

Use narrowing to replace ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) by

ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0))) 153.29/51.75
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.29/51.75
ACTIVE(g(a, x1)) → c5(G(mark(b), x1), ACTIVE(a)) 153.29/51.75
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.29/51.75
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
153.29/51.75
153.29/51.75

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 153.29/51.75
active(g(a, z0)) → mark(f(b, z0)) 153.29/51.75
active(f(z0, z0)) → mark(h(a)) 153.29/51.75
active(a) → mark(b) 153.29/51.75
active(h(z0)) → h(active(z0)) 153.29/51.75
active(g(z0, z1)) → g(active(z0), z1) 153.29/51.75
active(f(z0, z1)) → f(active(z0), z1) 153.29/51.75
h(mark(z0)) → mark(h(z0)) 153.29/51.75
h(ok(z0)) → ok(h(z0)) 153.29/51.75
g(mark(z0), z1) → mark(g(z0, z1)) 153.29/51.75
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 153.29/51.75
f(mark(z0), z1) → mark(f(z0, z1)) 153.29/51.75
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 153.29/51.75
proper(h(z0)) → h(proper(z0)) 153.29/51.75
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 153.29/51.75
proper(a) → ok(a) 153.29/51.75
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 153.29/51.75
proper(b) → ok(b) 153.29/51.75
top(mark(z0)) → top(proper(z0)) 153.29/51.75
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.75
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.75
H(mark(z0)) → c7(H(z0)) 153.29/51.75
H(ok(z0)) → c8(H(z0)) 153.29/51.75
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.75
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.75
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.75
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.75
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.75
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.75
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.75
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.75
ACTIVE(g(a, z0)) → c1 153.29/51.75
ACTIVE(f(z0, z0)) → c2 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0))) 153.29/51.75
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.75
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.75
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.29/51.75
ACTIVE(h(a)) → c4(H(mark(b))) 153.29/51.75
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0))) 153.29/51.75
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.29/51.75
ACTIVE(g(a, x1)) → c5(G(mark(b), x1), ACTIVE(a)) 153.29/51.75
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 153.29/51.75
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.29/51.75
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.77
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.77
H(mark(z0)) → c7(H(z0)) 153.29/51.77
H(ok(z0)) → c8(H(z0)) 153.29/51.77
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.77
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.77
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.77
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.77
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.77
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.77
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.77
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.77
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.77
ACTIVE(g(a, z0)) → c1 153.29/51.77
ACTIVE(f(z0, z0)) → c2 153.29/51.77
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0))) 153.29/51.77
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.77
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.77
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.29/51.77
ACTIVE(h(a)) → c4(H(mark(b))) 153.29/51.77
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0))) 153.29/51.77
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.29/51.77
ACTIVE(g(a, x1)) → c5(G(mark(b), x1), ACTIVE(a)) 153.29/51.77
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.29/51.77
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c1, c2, c4, c4, c5

153.29/51.77
153.29/51.77

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

Removed 1 trailing tuple parts
153.29/51.77
153.29/51.77

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 153.29/51.77
active(g(a, z0)) → mark(f(b, z0)) 153.29/51.77
active(f(z0, z0)) → mark(h(a)) 153.29/51.77
active(a) → mark(b) 153.29/51.77
active(h(z0)) → h(active(z0)) 153.29/51.77
active(g(z0, z1)) → g(active(z0), z1) 153.29/51.77
active(f(z0, z1)) → f(active(z0), z1) 153.29/51.77
h(mark(z0)) → mark(h(z0)) 153.29/51.77
h(ok(z0)) → ok(h(z0)) 153.29/51.77
g(mark(z0), z1) → mark(g(z0, z1)) 153.29/51.77
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 153.29/51.77
f(mark(z0), z1) → mark(f(z0, z1)) 153.29/51.77
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 153.29/51.77
proper(h(z0)) → h(proper(z0)) 153.29/51.77
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 153.29/51.77
proper(a) → ok(a) 153.29/51.77
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 153.29/51.77
proper(b) → ok(b) 153.29/51.77
top(mark(z0)) → top(proper(z0)) 153.29/51.77
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.77
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.77
H(mark(z0)) → c7(H(z0)) 153.29/51.77
H(ok(z0)) → c8(H(z0)) 153.29/51.77
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.77
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.77
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.77
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.77
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.77
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.77
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.77
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.77
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.77
ACTIVE(g(a, z0)) → c1 153.29/51.77
ACTIVE(f(z0, z0)) → c2 153.29/51.77
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0))) 153.29/51.77
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.77
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.77
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.29/51.77
ACTIVE(h(a)) → c4(H(mark(b))) 153.29/51.77
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0))) 153.29/51.77
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.29/51.77
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.29/51.77
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 153.29/51.77
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.77
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.77
H(mark(z0)) → c7(H(z0)) 153.29/51.77
H(ok(z0)) → c8(H(z0)) 153.29/51.77
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.77
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.77
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.77
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.77
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.77
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.77
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.77
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.77
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.77
ACTIVE(g(a, z0)) → c1 153.29/51.77
ACTIVE(f(z0, z0)) → c2 153.29/51.77
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0))) 153.29/51.77
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.77
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.77
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.29/51.77
ACTIVE(h(a)) → c4(H(mark(b))) 153.29/51.77
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0))) 153.29/51.77
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.29/51.77
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.29/51.77
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 153.29/51.77
ACTIVE(g(a, x1)) → c5(G(mark(b), x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c1, c2, c4, c4, c5, c5

153.29/51.77
153.29/51.77

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

Split RHS of tuples not part of any SCC
153.29/51.77
153.29/51.77

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 153.29/51.77
active(g(a, z0)) → mark(f(b, z0)) 153.29/51.77
active(f(z0, z0)) → mark(h(a)) 153.29/51.77
active(a) → mark(b) 153.29/51.77
active(h(z0)) → h(active(z0)) 153.29/51.77
active(g(z0, z1)) → g(active(z0), z1) 153.29/51.77
active(f(z0, z1)) → f(active(z0), z1) 153.29/51.77
h(mark(z0)) → mark(h(z0)) 153.29/51.77
h(ok(z0)) → ok(h(z0)) 153.29/51.77
g(mark(z0), z1) → mark(g(z0, z1)) 153.29/51.77
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 153.29/51.77
f(mark(z0), z1) → mark(f(z0, z1)) 153.29/51.77
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 153.29/51.77
proper(h(z0)) → h(proper(z0)) 153.29/51.77
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 153.29/51.77
proper(a) → ok(a) 153.29/51.77
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 153.29/51.77
proper(b) → ok(b) 153.29/51.77
top(mark(z0)) → top(proper(z0)) 153.29/51.77
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.77
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.77
H(mark(z0)) → c7(H(z0)) 153.29/51.77
H(ok(z0)) → c8(H(z0)) 153.29/51.77
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.77
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.77
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.77
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.77
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.77
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.77
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.77
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.77
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.77
ACTIVE(g(a, z0)) → c1 153.29/51.77
ACTIVE(f(z0, z0)) → c2 153.29/51.77
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.77
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.77
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.29/51.77
ACTIVE(h(a)) → c4(H(mark(b))) 153.29/51.77
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.29/51.77
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.29/51.77
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 153.29/51.77
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 153.29/51.77
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 153.29/51.77
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 153.29/51.77
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 153.29/51.77
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.77
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.77
H(mark(z0)) → c7(H(z0)) 153.29/51.77
H(ok(z0)) → c8(H(z0)) 153.29/51.77
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.77
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.77
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.77
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.77
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.77
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.77
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.77
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.77
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.77
ACTIVE(g(a, z0)) → c1 153.29/51.77
ACTIVE(f(z0, z0)) → c2 153.29/51.77
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.77
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.77
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.29/51.77
ACTIVE(h(a)) → c4(H(mark(b))) 153.29/51.77
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.29/51.77
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.29/51.77
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 153.29/51.77
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 153.29/51.77
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 153.29/51.77
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 153.29/51.77
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 153.29/51.77
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c1, c2, c4, c4, c5, c5, c3

153.29/51.77
153.29/51.77

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

Removed 2 trailing nodes:

ACTIVE(g(a, z0)) → c1 153.29/51.77
ACTIVE(f(z0, z0)) → c2
153.29/51.77
153.29/51.77

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 153.29/51.77
active(g(a, z0)) → mark(f(b, z0)) 153.29/51.77
active(f(z0, z0)) → mark(h(a)) 153.29/51.77
active(a) → mark(b) 153.29/51.77
active(h(z0)) → h(active(z0)) 153.29/51.77
active(g(z0, z1)) → g(active(z0), z1) 153.29/51.77
active(f(z0, z1)) → f(active(z0), z1) 153.29/51.77
h(mark(z0)) → mark(h(z0)) 153.29/51.77
h(ok(z0)) → ok(h(z0)) 153.29/51.77
g(mark(z0), z1) → mark(g(z0, z1)) 153.29/51.77
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 153.29/51.77
f(mark(z0), z1) → mark(f(z0, z1)) 153.29/51.77
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 153.29/51.77
proper(h(z0)) → h(proper(z0)) 153.29/51.77
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 153.29/51.77
proper(a) → ok(a) 153.29/51.77
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 153.29/51.77
proper(b) → ok(b) 153.29/51.77
top(mark(z0)) → top(proper(z0)) 153.29/51.77
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.77
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.77
H(mark(z0)) → c7(H(z0)) 153.29/51.77
H(ok(z0)) → c8(H(z0)) 153.29/51.77
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.77
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.77
F(mark(z0), z1) → c11(F(z0, z1)) 153.29/51.77
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.29/51.77
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.29/51.77
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.77
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.29/51.77
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.29/51.77
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.29/51.77
ACTIVE(g(a, z0)) → c1 153.29/51.77
ACTIVE(f(z0, z0)) → c2 153.29/51.77
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.29/51.77
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.29/51.77
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.29/51.77
ACTIVE(h(a)) → c4(H(mark(b))) 153.29/51.77
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.29/51.77
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 153.29/51.77
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.29/51.77
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 153.29/51.77
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 153.29/51.77
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 153.29/51.77
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 153.29/51.77
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 153.29/51.77
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.29/51.77
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) 153.29/51.77
H(mark(z0)) → c7(H(z0)) 153.29/51.77
H(ok(z0)) → c8(H(z0)) 153.29/51.77
G(mark(z0), z1) → c9(G(z0, z1)) 153.29/51.77
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.29/51.77
F(mark(z0), z1) → c11(F(z0, z1)) 153.59/51.81
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.59/51.81
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.59/51.81
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.59/51.81
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.59/51.81
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.59/51.81
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.59/51.81
ACTIVE(g(a, z0)) → c1 153.59/51.81
ACTIVE(f(z0, z0)) → c2 153.59/51.81
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.59/51.81
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.59/51.81
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.59/51.81
ACTIVE(h(a)) → c4(H(mark(b))) 153.59/51.81
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.59/51.81
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.59/51.81
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 153.59/51.81
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 153.59/51.81
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 153.59/51.81
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 153.59/51.81
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 153.59/51.81
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c1, c2, c4, c4, c5, c5, c3

153.59/51.81
153.59/51.81

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

Use narrowing to replace ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) by

ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0))) 153.59/51.81
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.59/51.81
ACTIVE(f(a, x1)) → c6(F(mark(b), x1), ACTIVE(a)) 153.59/51.81
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.59/51.81
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
153.59/51.81
153.59/51.81

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 153.59/51.81
active(g(a, z0)) → mark(f(b, z0)) 153.59/51.81
active(f(z0, z0)) → mark(h(a)) 153.59/51.81
active(a) → mark(b) 153.59/51.81
active(h(z0)) → h(active(z0)) 153.59/51.81
active(g(z0, z1)) → g(active(z0), z1) 153.59/51.81
active(f(z0, z1)) → f(active(z0), z1) 153.59/51.81
h(mark(z0)) → mark(h(z0)) 153.59/51.81
h(ok(z0)) → ok(h(z0)) 153.59/51.81
g(mark(z0), z1) → mark(g(z0, z1)) 153.59/51.81
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 153.59/51.81
f(mark(z0), z1) → mark(f(z0, z1)) 153.59/51.81
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 153.59/51.81
proper(h(z0)) → h(proper(z0)) 153.59/51.81
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 153.59/51.81
proper(a) → ok(a) 153.59/51.81
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 153.59/51.81
proper(b) → ok(b) 153.59/51.81
top(mark(z0)) → top(proper(z0)) 153.59/51.81
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.59/51.81
H(mark(z0)) → c7(H(z0)) 153.59/51.81
H(ok(z0)) → c8(H(z0)) 153.59/51.81
G(mark(z0), z1) → c9(G(z0, z1)) 153.59/51.81
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.59/51.81
F(mark(z0), z1) → c11(F(z0, z1)) 153.59/51.81
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.59/51.81
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.59/51.81
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.59/51.81
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.59/51.81
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.59/51.81
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.59/51.81
ACTIVE(g(a, z0)) → c1 153.59/51.81
ACTIVE(f(z0, z0)) → c2 153.59/51.81
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.59/51.81
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.59/51.81
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.59/51.81
ACTIVE(h(a)) → c4(H(mark(b))) 153.59/51.81
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.59/51.81
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.59/51.81
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 153.59/51.81
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 153.59/51.81
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 153.59/51.81
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 153.59/51.81
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 153.59/51.81
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 153.59/51.81
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0))) 153.59/51.81
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.59/51.81
ACTIVE(f(a, x1)) → c6(F(mark(b), x1), ACTIVE(a)) 153.59/51.81
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.59/51.81
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.59/51.81
H(mark(z0)) → c7(H(z0)) 153.59/51.81
H(ok(z0)) → c8(H(z0)) 153.59/51.81
G(mark(z0), z1) → c9(G(z0, z1)) 153.59/51.81
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.59/51.81
F(mark(z0), z1) → c11(F(z0, z1)) 153.59/51.81
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.59/51.81
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.59/51.81
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.59/51.81
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.59/51.81
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.59/51.81
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.59/51.81
ACTIVE(g(a, z0)) → c1 153.59/51.81
ACTIVE(f(z0, z0)) → c2 153.59/51.81
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.59/51.81
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.59/51.81
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.59/51.86
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.59/51.86
ACTIVE(h(a)) → c4(H(mark(b))) 153.59/51.86
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.59/51.86
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.59/51.86
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 153.59/51.86
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.59/51.86
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 153.59/51.86
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 153.59/51.86
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 153.59/51.86
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 153.59/51.86
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 153.59/51.86
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 153.59/51.86
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.59/51.86
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0))) 153.59/51.86
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.59/51.86
ACTIVE(f(a, x1)) → c6(F(mark(b), x1), ACTIVE(a)) 153.59/51.86
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 153.59/51.86
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.59/51.86
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c1, c2, c4, c4, c5, c5, c3, c6

153.59/51.86
153.59/51.86

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

Removed 1 trailing tuple parts
153.59/51.86
153.59/51.86

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 153.59/51.86
active(g(a, z0)) → mark(f(b, z0)) 153.59/51.86
active(f(z0, z0)) → mark(h(a)) 153.59/51.86
active(a) → mark(b) 153.59/51.86
active(h(z0)) → h(active(z0)) 153.59/51.86
active(g(z0, z1)) → g(active(z0), z1) 153.59/51.86
active(f(z0, z1)) → f(active(z0), z1) 153.59/51.86
h(mark(z0)) → mark(h(z0)) 153.59/51.86
h(ok(z0)) → ok(h(z0)) 153.59/51.86
g(mark(z0), z1) → mark(g(z0, z1)) 153.59/51.86
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 153.59/51.86
f(mark(z0), z1) → mark(f(z0, z1)) 153.59/51.86
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 153.59/51.86
proper(h(z0)) → h(proper(z0)) 153.59/51.86
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 153.59/51.86
proper(a) → ok(a) 153.59/51.86
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 153.59/51.86
proper(b) → ok(b) 153.59/51.86
top(mark(z0)) → top(proper(z0)) 153.59/51.86
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.59/51.86
H(mark(z0)) → c7(H(z0)) 153.59/51.86
H(ok(z0)) → c8(H(z0)) 153.59/51.86
G(mark(z0), z1) → c9(G(z0, z1)) 153.59/51.86
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.59/51.86
F(mark(z0), z1) → c11(F(z0, z1)) 153.59/51.86
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.59/51.86
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.59/51.86
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.59/51.86
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 153.59/51.86
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 153.59/51.86
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 153.59/51.86
ACTIVE(g(a, z0)) → c1 153.59/51.86
ACTIVE(f(z0, z0)) → c2 153.59/51.86
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 153.59/51.86
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 153.59/51.86
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 153.59/51.86
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 153.59/51.86
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 153.59/51.86
ACTIVE(h(a)) → c4(H(mark(b))) 153.59/51.86
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.59/51.86
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.59/51.86
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 153.59/51.86
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.59/51.86
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 153.59/51.86
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 153.59/51.86
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 153.59/51.86
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 153.59/51.86
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 153.59/51.86
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 153.59/51.86
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 153.59/51.86
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0))) 153.59/51.86
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 153.59/51.86
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 153.59/51.86
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 153.59/51.86
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 153.59/51.86
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 153.59/51.86
H(mark(z0)) → c7(H(z0)) 153.59/51.86
H(ok(z0)) → c8(H(z0)) 153.59/51.86
G(mark(z0), z1) → c9(G(z0, z1)) 153.59/51.86
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 153.59/51.86
F(mark(z0), z1) → c11(F(z0, z1)) 153.59/51.86
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 153.59/51.86
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 153.59/51.86
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.96
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.96
ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(f(a, x1)) → c6(F(mark(b), x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6

154.01/51.96
154.01/51.96

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

Split RHS of tuples not part of any SCC
154.01/51.96
154.01/51.96

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.01/51.96
active(g(a, z0)) → mark(f(b, z0)) 154.01/51.96
active(f(z0, z0)) → mark(h(a)) 154.01/51.96
active(a) → mark(b) 154.01/51.96
active(h(z0)) → h(active(z0)) 154.01/51.96
active(g(z0, z1)) → g(active(z0), z1) 154.01/51.96
active(f(z0, z1)) → f(active(z0), z1) 154.01/51.96
h(mark(z0)) → mark(h(z0)) 154.01/51.96
h(ok(z0)) → ok(h(z0)) 154.01/51.96
g(mark(z0), z1) → mark(g(z0, z1)) 154.01/51.96
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.01/51.96
f(mark(z0), z1) → mark(f(z0, z1)) 154.01/51.96
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.01/51.96
proper(h(z0)) → h(proper(z0)) 154.01/51.96
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.01/51.96
proper(a) → ok(a) 154.01/51.96
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.01/51.96
proper(b) → ok(b) 154.01/51.96
top(mark(z0)) → top(proper(z0)) 154.01/51.96
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.96
H(mark(z0)) → c7(H(z0)) 154.01/51.96
H(ok(z0)) → c8(H(z0)) 154.01/51.96
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.96
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.96
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.96
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.96
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 154.01/51.96
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.96
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.96
ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.96
H(mark(z0)) → c7(H(z0)) 154.01/51.96
H(ok(z0)) → c8(H(z0)) 154.01/51.96
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.96
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.96
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.96
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.96
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 154.01/51.96
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.96
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.96
ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15

154.01/51.96
154.01/51.96

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

Removed 2 trailing nodes:

ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2
154.01/51.96
154.01/51.96

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.01/51.96
active(g(a, z0)) → mark(f(b, z0)) 154.01/51.96
active(f(z0, z0)) → mark(h(a)) 154.01/51.96
active(a) → mark(b) 154.01/51.96
active(h(z0)) → h(active(z0)) 154.01/51.96
active(g(z0, z1)) → g(active(z0), z1) 154.01/51.96
active(f(z0, z1)) → f(active(z0), z1) 154.01/51.96
h(mark(z0)) → mark(h(z0)) 154.01/51.96
h(ok(z0)) → ok(h(z0)) 154.01/51.96
g(mark(z0), z1) → mark(g(z0, z1)) 154.01/51.96
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.01/51.96
f(mark(z0), z1) → mark(f(z0, z1)) 154.01/51.96
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.01/51.96
proper(h(z0)) → h(proper(z0)) 154.01/51.96
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.01/51.96
proper(a) → ok(a) 154.01/51.96
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.01/51.96
proper(b) → ok(b) 154.01/51.96
top(mark(z0)) → top(proper(z0)) 154.01/51.96
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.96
H(mark(z0)) → c7(H(z0)) 154.01/51.96
H(ok(z0)) → c8(H(z0)) 154.01/51.96
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.96
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.96
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.96
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.96
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 154.01/51.96
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.96
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.96
ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.96
H(mark(z0)) → c7(H(z0)) 154.01/51.96
H(ok(z0)) → c8(H(z0)) 154.01/51.96
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.96
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.96
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.96
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.96
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) 154.01/51.96
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.96
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.96
ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15

154.01/51.96
154.01/51.96

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

Use narrowing to replace PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) by

PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.01/51.96
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.01/51.96
PROPER(h(a)) → c13(H(ok(a)), PROPER(a)) 154.01/51.96
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.01/51.96
PROPER(h(b)) → c13(H(ok(b)), PROPER(b))
154.01/51.96
154.01/51.96

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.01/51.96
active(g(a, z0)) → mark(f(b, z0)) 154.01/51.96
active(f(z0, z0)) → mark(h(a)) 154.01/51.96
active(a) → mark(b) 154.01/51.96
active(h(z0)) → h(active(z0)) 154.01/51.96
active(g(z0, z1)) → g(active(z0), z1) 154.01/51.96
active(f(z0, z1)) → f(active(z0), z1) 154.01/51.96
h(mark(z0)) → mark(h(z0)) 154.01/51.96
h(ok(z0)) → ok(h(z0)) 154.01/51.96
g(mark(z0), z1) → mark(g(z0, z1)) 154.01/51.96
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.01/51.96
f(mark(z0), z1) → mark(f(z0, z1)) 154.01/51.96
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.01/51.96
proper(h(z0)) → h(proper(z0)) 154.01/51.96
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.01/51.96
proper(a) → ok(a) 154.01/51.96
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.01/51.96
proper(b) → ok(b) 154.01/51.96
top(mark(z0)) → top(proper(z0)) 154.01/51.96
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.96
H(mark(z0)) → c7(H(z0)) 154.01/51.96
H(ok(z0)) → c8(H(z0)) 154.01/51.96
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.96
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.96
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.96
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.96
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.96
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.96
ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.01/51.96
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.01/51.96
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.01/51.96
PROPER(h(a)) → c13(H(ok(a)), PROPER(a)) 154.01/51.96
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.01/51.96
PROPER(h(b)) → c13(H(ok(b)), PROPER(b))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.96
H(mark(z0)) → c7(H(z0)) 154.01/51.96
H(ok(z0)) → c8(H(z0)) 154.01/51.96
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.96
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.96
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.96
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.96
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.96
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.96
ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.01/51.96
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.01/51.96
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.01/51.96
PROPER(h(a)) → c13(H(ok(a)), PROPER(a)) 154.01/51.96
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.01/51.96
PROPER(h(b)) → c13(H(ok(b)), PROPER(b))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c14, c16, c18, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13

154.01/51.96
154.01/51.96

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

Removed 2 trailing tuple parts
154.01/51.96
154.01/51.96

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.01/51.96
active(g(a, z0)) → mark(f(b, z0)) 154.01/51.96
active(f(z0, z0)) → mark(h(a)) 154.01/51.96
active(a) → mark(b) 154.01/51.96
active(h(z0)) → h(active(z0)) 154.01/51.96
active(g(z0, z1)) → g(active(z0), z1) 154.01/51.96
active(f(z0, z1)) → f(active(z0), z1) 154.01/51.96
h(mark(z0)) → mark(h(z0)) 154.01/51.96
h(ok(z0)) → ok(h(z0)) 154.01/51.96
g(mark(z0), z1) → mark(g(z0, z1)) 154.01/51.96
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.01/51.96
f(mark(z0), z1) → mark(f(z0, z1)) 154.01/51.96
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.01/51.96
proper(h(z0)) → h(proper(z0)) 154.01/51.96
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.01/51.96
proper(a) → ok(a) 154.01/51.96
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.01/51.96
proper(b) → ok(b) 154.01/51.96
top(mark(z0)) → top(proper(z0)) 154.01/51.96
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.96
H(mark(z0)) → c7(H(z0)) 154.01/51.96
H(ok(z0)) → c8(H(z0)) 154.01/51.96
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.96
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.96
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.96
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.96
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.96
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.96
ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.01/51.96
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.01/51.96
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.01/51.96
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.01/51.96
PROPER(h(a)) → c13(H(ok(a))) 154.01/51.96
PROPER(h(b)) → c13(H(ok(b)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.96
H(mark(z0)) → c7(H(z0)) 154.01/51.96
H(ok(z0)) → c8(H(z0)) 154.01/51.96
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.96
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.96
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.96
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.96
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.96
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.96
ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.01/51.96
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.01/51.96
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.01/51.96
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.01/51.96
PROPER(h(a)) → c13(H(ok(a))) 154.01/51.96
PROPER(h(b)) → c13(H(ok(b)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c14, c16, c18, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13

154.01/51.96
154.01/51.96

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

Removed 2 trailing nodes:

ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2
154.01/51.96
154.01/51.96

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.01/51.96
active(g(a, z0)) → mark(f(b, z0)) 154.01/51.96
active(f(z0, z0)) → mark(h(a)) 154.01/51.96
active(a) → mark(b) 154.01/51.96
active(h(z0)) → h(active(z0)) 154.01/51.96
active(g(z0, z1)) → g(active(z0), z1) 154.01/51.96
active(f(z0, z1)) → f(active(z0), z1) 154.01/51.96
h(mark(z0)) → mark(h(z0)) 154.01/51.96
h(ok(z0)) → ok(h(z0)) 154.01/51.96
g(mark(z0), z1) → mark(g(z0, z1)) 154.01/51.96
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.01/51.96
f(mark(z0), z1) → mark(f(z0, z1)) 154.01/51.96
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.01/51.96
proper(h(z0)) → h(proper(z0)) 154.01/51.96
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.01/51.96
proper(a) → ok(a) 154.01/51.96
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.01/51.96
proper(b) → ok(b) 154.01/51.96
top(mark(z0)) → top(proper(z0)) 154.01/51.96
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.96
H(mark(z0)) → c7(H(z0)) 154.01/51.96
H(ok(z0)) → c8(H(z0)) 154.01/51.96
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.96
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.96
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.96
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.96
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.96
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.96
ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.01/51.96
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.01/51.96
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.01/51.96
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.01/51.96
PROPER(h(a)) → c13(H(ok(a))) 154.01/51.96
PROPER(h(b)) → c13(H(ok(b)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.96
H(mark(z0)) → c7(H(z0)) 154.01/51.96
H(ok(z0)) → c8(H(z0)) 154.01/51.96
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.96
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.96
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.96
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.96
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.96
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.96
ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.01/51.96
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.01/51.96
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.01/51.96
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.01/51.96
PROPER(h(a)) → c13(H(ok(a))) 154.01/51.96
PROPER(h(b)) → c13(H(ok(b)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c14, c16, c18, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13

154.01/51.96
154.01/51.96

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

Use narrowing to replace PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.01/51.96
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.01/51.96
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0), PROPER(a)) 154.01/51.96
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.01/51.96
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0), PROPER(b)) 154.01/51.96
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.01/51.96
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.01/51.96
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(a), PROPER(x1)) 154.01/51.96
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.01/51.96
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(b), PROPER(x1))
154.01/51.96
154.01/51.96

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.01/51.96
active(g(a, z0)) → mark(f(b, z0)) 154.01/51.96
active(f(z0, z0)) → mark(h(a)) 154.01/51.96
active(a) → mark(b) 154.01/51.96
active(h(z0)) → h(active(z0)) 154.01/51.96
active(g(z0, z1)) → g(active(z0), z1) 154.01/51.96
active(f(z0, z1)) → f(active(z0), z1) 154.01/51.96
h(mark(z0)) → mark(h(z0)) 154.01/51.96
h(ok(z0)) → ok(h(z0)) 154.01/51.96
g(mark(z0), z1) → mark(g(z0, z1)) 154.01/51.96
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.01/51.96
f(mark(z0), z1) → mark(f(z0, z1)) 154.01/51.96
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.01/51.96
proper(h(z0)) → h(proper(z0)) 154.01/51.96
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.01/51.96
proper(a) → ok(a) 154.01/51.96
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.01/51.96
proper(b) → ok(b) 154.01/51.96
top(mark(z0)) → top(proper(z0)) 154.01/51.96
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.96
H(mark(z0)) → c7(H(z0)) 154.01/51.96
H(ok(z0)) → c8(H(z0)) 154.01/51.96
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.96
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.96
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.96
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.96
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.96
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.96
ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.01/51.96
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.01/51.96
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.01/51.96
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.01/51.96
PROPER(h(a)) → c13(H(ok(a))) 154.01/51.96
PROPER(h(b)) → c13(H(ok(b))) 154.01/51.96
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.01/51.96
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.01/51.96
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0), PROPER(a)) 154.01/51.96
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.01/51.96
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0), PROPER(b)) 154.01/51.96
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.01/51.96
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.01/51.96
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(a), PROPER(x1)) 154.01/51.96
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.01/51.96
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(b), PROPER(x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.96
H(mark(z0)) → c7(H(z0)) 154.01/51.96
H(ok(z0)) → c8(H(z0)) 154.01/51.96
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.96
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.96
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.96
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.96
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.96
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.96
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.96
ACTIVE(g(a, z0)) → c1 154.01/51.96
ACTIVE(f(z0, z0)) → c2 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.96
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.96
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.96
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.96
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.96
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.96
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.01/51.96
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.01/51.96
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.01/51.96
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.01/51.96
PROPER(h(a)) → c13(H(ok(a))) 154.01/51.96
PROPER(h(b)) → c13(H(ok(b))) 154.01/51.96
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.01/51.96
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.01/51.96
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0), PROPER(a)) 154.01/51.97
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.01/51.97
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0), PROPER(b)) 154.01/51.97
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.01/51.97
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.01/51.97
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(a), PROPER(x1)) 154.01/51.97
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.01/51.97
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(b), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c16, c18, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13, c14

154.01/51.97
154.01/51.97

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

Removed 4 trailing tuple parts
154.01/51.97
154.01/51.97

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.01/51.97
active(g(a, z0)) → mark(f(b, z0)) 154.01/51.97
active(f(z0, z0)) → mark(h(a)) 154.01/51.97
active(a) → mark(b) 154.01/51.97
active(h(z0)) → h(active(z0)) 154.01/51.97
active(g(z0, z1)) → g(active(z0), z1) 154.01/51.97
active(f(z0, z1)) → f(active(z0), z1) 154.01/51.97
h(mark(z0)) → mark(h(z0)) 154.01/51.97
h(ok(z0)) → ok(h(z0)) 154.01/51.97
g(mark(z0), z1) → mark(g(z0, z1)) 154.01/51.97
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.01/51.97
f(mark(z0), z1) → mark(f(z0, z1)) 154.01/51.97
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.01/51.97
proper(h(z0)) → h(proper(z0)) 154.01/51.97
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.01/51.97
proper(a) → ok(a) 154.01/51.97
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.01/51.97
proper(b) → ok(b) 154.01/51.97
top(mark(z0)) → top(proper(z0)) 154.01/51.97
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.97
H(mark(z0)) → c7(H(z0)) 154.01/51.97
H(ok(z0)) → c8(H(z0)) 154.01/51.97
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.97
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.97
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.97
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.97
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.97
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.97
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.97
ACTIVE(g(a, z0)) → c1 154.01/51.97
ACTIVE(f(z0, z0)) → c2 154.01/51.97
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.97
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.97
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.97
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.97
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.97
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.97
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.97
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.97
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.97
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.97
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.97
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.97
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.97
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.97
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.97
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.97
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.97
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.01/51.97
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.01/51.97
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.01/51.97
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.01/51.97
PROPER(h(a)) → c13(H(ok(a))) 154.01/51.97
PROPER(h(b)) → c13(H(ok(b))) 154.01/51.97
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.01/51.97
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.01/51.97
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.01/51.97
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.01/51.97
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.01/51.97
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.01/51.97
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.01/51.97
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.01/51.97
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.01/51.97
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.97
H(mark(z0)) → c7(H(z0)) 154.01/51.97
H(ok(z0)) → c8(H(z0)) 154.01/51.97
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.97
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.97
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.97
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.97
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.97
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.97
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.97
ACTIVE(g(a, z0)) → c1 154.01/51.97
ACTIVE(f(z0, z0)) → c2 154.01/51.97
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.97
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.97
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.97
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.97
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.97
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.97
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.97
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.97
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.97
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.97
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.97
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.97
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.97
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.97
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.97
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.97
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.97
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.01/51.97
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.01/51.97
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.01/51.97
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.01/51.97
PROPER(h(a)) → c13(H(ok(a))) 154.01/51.97
PROPER(h(b)) → c13(H(ok(b))) 154.01/51.97
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.01/51.97
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.01/51.97
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.01/51.97
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.01/51.97
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.01/51.97
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.01/51.97
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.01/51.97
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.01/51.97
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.01/51.97
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c16, c18, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13, c14, c14

154.01/51.97
154.01/51.97

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

Removed 2 trailing nodes:

ACTIVE(g(a, z0)) → c1 154.01/51.97
ACTIVE(f(z0, z0)) → c2
154.01/51.97
154.01/51.97

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.01/51.97
active(g(a, z0)) → mark(f(b, z0)) 154.01/51.97
active(f(z0, z0)) → mark(h(a)) 154.01/51.97
active(a) → mark(b) 154.01/51.97
active(h(z0)) → h(active(z0)) 154.01/51.97
active(g(z0, z1)) → g(active(z0), z1) 154.01/51.97
active(f(z0, z1)) → f(active(z0), z1) 154.01/51.97
h(mark(z0)) → mark(h(z0)) 154.01/51.97
h(ok(z0)) → ok(h(z0)) 154.01/51.97
g(mark(z0), z1) → mark(g(z0, z1)) 154.01/51.97
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.01/51.97
f(mark(z0), z1) → mark(f(z0, z1)) 154.01/51.97
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.01/51.97
proper(h(z0)) → h(proper(z0)) 154.01/51.97
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.01/51.97
proper(a) → ok(a) 154.01/51.97
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.01/51.97
proper(b) → ok(b) 154.01/51.97
top(mark(z0)) → top(proper(z0)) 154.01/51.97
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.97
H(mark(z0)) → c7(H(z0)) 154.01/51.97
H(ok(z0)) → c8(H(z0)) 154.01/51.97
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.97
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.97
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.97
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.97
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.97
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.97
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.97
ACTIVE(g(a, z0)) → c1 154.01/51.97
ACTIVE(f(z0, z0)) → c2 154.01/51.97
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.97
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.97
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.97
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.97
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.97
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.97
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.97
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.97
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.97
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.97
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.97
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.97
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.97
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.97
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.97
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.97
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.97
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.01/51.97
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.01/51.97
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.01/51.97
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.01/51.97
PROPER(h(a)) → c13(H(ok(a))) 154.01/51.97
PROPER(h(b)) → c13(H(ok(b))) 154.01/51.97
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.01/51.97
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.01/51.97
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.01/51.97
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.01/51.97
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.01/51.97
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.01/51.97
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.01/51.97
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.01/51.97
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.01/51.97
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.01/51.97
H(mark(z0)) → c7(H(z0)) 154.01/51.97
H(ok(z0)) → c8(H(z0)) 154.01/51.97
G(mark(z0), z1) → c9(G(z0, z1)) 154.01/51.97
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.01/51.97
F(mark(z0), z1) → c11(F(z0, z1)) 154.01/51.97
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.01/51.97
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 154.01/51.97
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.01/51.97
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.01/51.97
ACTIVE(g(a, z0)) → c1 154.01/51.97
ACTIVE(f(z0, z0)) → c2 154.01/51.97
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.01/51.97
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.01/51.97
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.01/51.97
ACTIVE(h(a)) → c4(H(mark(b))) 154.01/51.97
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.97
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.97
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.97
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.01/51.97
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.01/51.97
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.01/51.97
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.01/51.97
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.01/51.97
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.01/51.97
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.01/51.97
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.01/51.97
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.01/51.97
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.01/51.97
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.01/51.97
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.01/51.97
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c16, c18, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13, c14, c14

154.32/52.06
154.32/52.06

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

Use narrowing to replace PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0), PROPER(a)) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0), PROPER(b)) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(a), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
154.32/52.06
154.32/52.06

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.06
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.06
active(f(z0, z0)) → mark(h(a)) 154.32/52.06
active(a) → mark(b) 154.32/52.06
active(h(z0)) → h(active(z0)) 154.32/52.06
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.06
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.06
h(mark(z0)) → mark(h(z0)) 154.32/52.06
h(ok(z0)) → ok(h(z0)) 154.32/52.06
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.06
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.06
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.06
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.06
proper(h(z0)) → h(proper(z0)) 154.32/52.06
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.06
proper(a) → ok(a) 154.32/52.06
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.06
proper(b) → ok(b) 154.32/52.06
top(mark(z0)) → top(proper(z0)) 154.32/52.06
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.06
H(mark(z0)) → c7(H(z0)) 154.32/52.06
H(ok(z0)) → c8(H(z0)) 154.32/52.06
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.06
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.06
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.06
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.06
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.32/52.06
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.06
ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.06
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0), PROPER(a)) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0), PROPER(b)) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(a), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.06
H(mark(z0)) → c7(H(z0)) 154.32/52.06
H(ok(z0)) → c8(H(z0)) 154.32/52.06
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.06
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.06
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.06
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.06
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.32/52.06
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.06
ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.06
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0), PROPER(a)) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0), PROPER(b)) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(a), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c18, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13, c14, c14, c16

154.32/52.06
154.32/52.06

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

Removed 4 trailing tuple parts
154.32/52.06
154.32/52.06

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.06
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.06
active(f(z0, z0)) → mark(h(a)) 154.32/52.06
active(a) → mark(b) 154.32/52.06
active(h(z0)) → h(active(z0)) 154.32/52.06
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.06
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.06
h(mark(z0)) → mark(h(z0)) 154.32/52.06
h(ok(z0)) → ok(h(z0)) 154.32/52.06
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.06
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.06
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.06
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.06
proper(h(z0)) → h(proper(z0)) 154.32/52.06
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.06
proper(a) → ok(a) 154.32/52.06
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.06
proper(b) → ok(b) 154.32/52.06
top(mark(z0)) → top(proper(z0)) 154.32/52.06
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.06
H(mark(z0)) → c7(H(z0)) 154.32/52.06
H(ok(z0)) → c8(H(z0)) 154.32/52.06
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.06
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.06
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.06
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.06
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.32/52.06
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.06
ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.06
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.06
H(mark(z0)) → c7(H(z0)) 154.32/52.06
H(ok(z0)) → c8(H(z0)) 154.32/52.06
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.06
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.06
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.06
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.06
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.32/52.06
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.06
ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.06
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c18, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13, c14, c14, c16, c16

154.32/52.06
154.32/52.06

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

Removed 2 trailing nodes:

ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2
154.32/52.06
154.32/52.06

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.06
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.06
active(f(z0, z0)) → mark(h(a)) 154.32/52.06
active(a) → mark(b) 154.32/52.06
active(h(z0)) → h(active(z0)) 154.32/52.06
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.06
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.06
h(mark(z0)) → mark(h(z0)) 154.32/52.06
h(ok(z0)) → ok(h(z0)) 154.32/52.06
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.06
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.06
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.06
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.06
proper(h(z0)) → h(proper(z0)) 154.32/52.06
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.06
proper(a) → ok(a) 154.32/52.06
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.06
proper(b) → ok(b) 154.32/52.06
top(mark(z0)) → top(proper(z0)) 154.32/52.06
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.06
H(mark(z0)) → c7(H(z0)) 154.32/52.06
H(ok(z0)) → c8(H(z0)) 154.32/52.06
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.06
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.06
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.06
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.06
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.32/52.06
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.06
ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.06
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.06
H(mark(z0)) → c7(H(z0)) 154.32/52.06
H(ok(z0)) → c8(H(z0)) 154.32/52.06
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.06
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.06
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.06
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.06
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 154.32/52.06
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.06
ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.06
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c18, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13, c14, c14, c16, c16

154.32/52.06
154.32/52.06

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

Use narrowing to replace TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
TOP(mark(a)) → c18(TOP(ok(a)), PROPER(a)) 154.32/52.06
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
TOP(mark(b)) → c18(TOP(ok(b)), PROPER(b))
154.32/52.06
154.32/52.06

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.06
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.06
active(f(z0, z0)) → mark(h(a)) 154.32/52.06
active(a) → mark(b) 154.32/52.06
active(h(z0)) → h(active(z0)) 154.32/52.06
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.06
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.06
h(mark(z0)) → mark(h(z0)) 154.32/52.06
h(ok(z0)) → ok(h(z0)) 154.32/52.06
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.06
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.06
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.06
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.06
proper(h(z0)) → h(proper(z0)) 154.32/52.06
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.06
proper(a) → ok(a) 154.32/52.06
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.06
proper(b) → ok(b) 154.32/52.06
top(mark(z0)) → top(proper(z0)) 154.32/52.06
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.06
H(mark(z0)) → c7(H(z0)) 154.32/52.06
H(ok(z0)) → c8(H(z0)) 154.32/52.06
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.06
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.06
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.06
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.06
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.06
ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.06
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
TOP(mark(a)) → c18(TOP(ok(a)), PROPER(a)) 154.32/52.06
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
TOP(mark(b)) → c18(TOP(ok(b)), PROPER(b))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.06
H(mark(z0)) → c7(H(z0)) 154.32/52.06
H(ok(z0)) → c8(H(z0)) 154.32/52.06
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.06
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.06
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.06
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.06
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.06
ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.06
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
TOP(mark(a)) → c18(TOP(ok(a)), PROPER(a)) 154.32/52.06
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
TOP(mark(b)) → c18(TOP(ok(b)), PROPER(b))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13, c14, c14, c16, c16, c18

154.32/52.06
154.32/52.06

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

Removed 2 trailing tuple parts
154.32/52.06
154.32/52.06

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.06
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.06
active(f(z0, z0)) → mark(h(a)) 154.32/52.06
active(a) → mark(b) 154.32/52.06
active(h(z0)) → h(active(z0)) 154.32/52.06
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.06
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.06
h(mark(z0)) → mark(h(z0)) 154.32/52.06
h(ok(z0)) → ok(h(z0)) 154.32/52.06
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.06
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.06
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.06
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.06
proper(h(z0)) → h(proper(z0)) 154.32/52.06
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.06
proper(a) → ok(a) 154.32/52.06
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.06
proper(b) → ok(b) 154.32/52.06
top(mark(z0)) → top(proper(z0)) 154.32/52.06
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.06
H(mark(z0)) → c7(H(z0)) 154.32/52.06
H(ok(z0)) → c8(H(z0)) 154.32/52.06
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.06
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.06
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.06
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.06
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.06
ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.06
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
TOP(mark(a)) → c18(TOP(ok(a))) 154.32/52.06
TOP(mark(b)) → c18(TOP(ok(b)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.06
H(mark(z0)) → c7(H(z0)) 154.32/52.06
H(ok(z0)) → c8(H(z0)) 154.32/52.06
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.06
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.06
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.06
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.06
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.06
ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.06
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
TOP(mark(a)) → c18(TOP(ok(a))) 154.32/52.06
TOP(mark(b)) → c18(TOP(ok(b)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13, c14, c14, c16, c16, c18, c18

154.32/52.06
154.32/52.06

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

Removed 2 trailing nodes:

ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2
154.32/52.06
154.32/52.06

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.06
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.06
active(f(z0, z0)) → mark(h(a)) 154.32/52.06
active(a) → mark(b) 154.32/52.06
active(h(z0)) → h(active(z0)) 154.32/52.06
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.06
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.06
h(mark(z0)) → mark(h(z0)) 154.32/52.06
h(ok(z0)) → ok(h(z0)) 154.32/52.06
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.06
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.06
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.06
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.06
proper(h(z0)) → h(proper(z0)) 154.32/52.06
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.06
proper(a) → ok(a) 154.32/52.06
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.06
proper(b) → ok(b) 154.32/52.06
top(mark(z0)) → top(proper(z0)) 154.32/52.06
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.06
H(mark(z0)) → c7(H(z0)) 154.32/52.06
H(ok(z0)) → c8(H(z0)) 154.32/52.06
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.06
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.06
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.06
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.06
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.06
ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.06
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
TOP(mark(a)) → c18(TOP(ok(a))) 154.32/52.06
TOP(mark(b)) → c18(TOP(ok(b)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.06
H(mark(z0)) → c7(H(z0)) 154.32/52.06
H(ok(z0)) → c8(H(z0)) 154.32/52.06
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.06
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.06
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.06
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.06
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.06
ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.06
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
TOP(mark(a)) → c18(TOP(ok(a))) 154.32/52.06
TOP(mark(b)) → c18(TOP(ok(b)))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13, c14, c14, c16, c16, c18, c18

154.32/52.06
154.32/52.06

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

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

TOP(mark(b)) → c18(TOP(ok(b)))
We considered the (Usable) Rules:

proper(h(z0)) → h(proper(z0)) 154.32/52.06
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.06
proper(a) → ok(a) 154.32/52.06
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.06
proper(b) → ok(b) 154.32/52.06
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.06
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.06
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.06
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.06
h(mark(z0)) → mark(h(z0)) 154.32/52.06
h(ok(z0)) → ok(h(z0)) 154.32/52.06
active(h(z0)) → mark(g(z0, z0)) 154.32/52.06
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.06
active(f(z0, z0)) → mark(h(a)) 154.32/52.06
active(a) → mark(b) 154.32/52.06
active(h(z0)) → h(active(z0)) 154.32/52.06
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.06
active(f(z0, z1)) → f(active(z0), z1)
And the Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.06
H(mark(z0)) → c7(H(z0)) 154.32/52.06
H(ok(z0)) → c8(H(z0)) 154.32/52.06
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.06
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.06
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.06
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.06
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.06
ACTIVE(g(a, z0)) → c1 154.32/52.06
ACTIVE(f(z0, z0)) → c2 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.06
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.06
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.06
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.06
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.06
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.06
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.06
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.06
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.06
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.06
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.06
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.06
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.06
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.06
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.06
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.06
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.06
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.06
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.06
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.06
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.08
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.08
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.08
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.08
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.08
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.08
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.08
TOP(mark(a)) → c18(TOP(ok(a))) 154.32/52.08
TOP(mark(b)) → c18(TOP(ok(b)))
The order we found is given by the following interpretation:
Polynomial interpretation : 154.32/52.08

POL(ACTIVE(x1)) = 0    154.32/52.08
POL(F(x1, x2)) = 0    154.32/52.08
POL(G(x1, x2)) = 0    154.32/52.08
POL(H(x1)) = 0    154.32/52.08
POL(PROPER(x1)) = 0    154.32/52.08
POL(TOP(x1)) = x1    154.32/52.08
POL(a) = [4]    154.32/52.08
POL(active(x1)) = x1    154.32/52.08
POL(b) = 0    154.32/52.08
POL(c(x1)) = x1    154.32/52.08
POL(c1) = 0    154.32/52.08
POL(c10(x1)) = x1    154.32/52.08
POL(c11(x1)) = x1    154.32/52.08
POL(c12(x1)) = x1    154.32/52.08
POL(c13(x1)) = x1    154.32/52.08
POL(c13(x1, x2)) = x1 + x2    154.32/52.08
POL(c14(x1, x2)) = x1 + x2    154.32/52.08
POL(c14(x1, x2, x3)) = x1 + x2 + x3    154.32/52.08
POL(c15(x1)) = x1    154.32/52.08
POL(c16(x1, x2)) = x1 + x2    154.32/52.08
POL(c16(x1, x2, x3)) = x1 + x2 + x3    154.32/52.08
POL(c18(x1)) = x1    154.32/52.08
POL(c18(x1, x2)) = x1 + x2    154.32/52.08
POL(c19(x1, x2)) = x1 + x2    154.32/52.08
POL(c2) = 0    154.32/52.08
POL(c3(x1)) = x1    154.32/52.08
POL(c4(x1)) = x1    154.32/52.08
POL(c4(x1, x2)) = x1 + x2    154.32/52.08
POL(c5(x1)) = x1    154.32/52.08
POL(c5(x1, x2)) = x1 + x2    154.32/52.08
POL(c6(x1)) = x1    154.32/52.08
POL(c6(x1, x2)) = x1 + x2    154.32/52.08
POL(c7(x1)) = x1    154.32/52.08
POL(c8(x1)) = x1    154.32/52.08
POL(c9(x1)) = x1    154.32/52.08
POL(f(x1, x2)) = [4]    154.32/52.08
POL(g(x1, x2)) = [4]    154.32/52.08
POL(h(x1)) = [4]    154.32/52.08
POL(mark(x1)) = [4]    154.32/52.08
POL(ok(x1)) = x1    154.32/52.08
POL(proper(x1)) = 0   
154.32/52.08
154.32/52.08

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.08
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.08
active(f(z0, z0)) → mark(h(a)) 154.32/52.08
active(a) → mark(b) 154.32/52.08
active(h(z0)) → h(active(z0)) 154.32/52.08
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.08
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.08
h(mark(z0)) → mark(h(z0)) 154.32/52.08
h(ok(z0)) → ok(h(z0)) 154.32/52.08
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.08
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.08
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.08
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.08
proper(h(z0)) → h(proper(z0)) 154.32/52.08
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.08
proper(a) → ok(a) 154.32/52.08
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.08
proper(b) → ok(b) 154.32/52.08
top(mark(z0)) → top(proper(z0)) 154.32/52.08
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.08
H(mark(z0)) → c7(H(z0)) 154.32/52.08
H(ok(z0)) → c8(H(z0)) 154.32/52.08
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.08
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.08
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.08
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.08
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.08
ACTIVE(g(a, z0)) → c1 154.32/52.08
ACTIVE(f(z0, z0)) → c2 154.32/52.08
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.08
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.08
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.08
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.08
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.08
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.08
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.08
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.08
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.08
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.08
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.08
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.08
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.08
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.08
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.08
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.08
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.08
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.08
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.08
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.08
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.08
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.08
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.08
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.08
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.08
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.08
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.08
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.08
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.08
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.08
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.08
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.08
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.08
TOP(mark(a)) → c18(TOP(ok(a))) 154.32/52.08
TOP(mark(b)) → c18(TOP(ok(b)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.08
H(mark(z0)) → c7(H(z0)) 154.32/52.08
H(ok(z0)) → c8(H(z0)) 154.32/52.08
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.08
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.08
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.08
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.08
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.08
ACTIVE(g(a, z0)) → c1 154.32/52.08
ACTIVE(f(z0, z0)) → c2 154.32/52.08
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.08
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.08
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.08
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.08
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.08
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.08
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.08
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.08
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.08
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.08
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.08
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.08
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.08
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.08
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.08
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.08
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.08
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.08
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.08
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.08
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.08
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.08
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.08
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.08
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.08
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.08
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.08
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.08
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.08
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.08
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.08
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.08
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.08
TOP(mark(a)) → c18(TOP(ok(a)))
K tuples:

TOP(mark(b)) → c18(TOP(ok(b)))
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13, c14, c14, c16, c16, c18, c18

154.32/52.08
154.32/52.08

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

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

TOP(mark(a)) → c18(TOP(ok(a)))
We considered the (Usable) Rules:

proper(h(z0)) → h(proper(z0)) 154.32/52.08
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.08
proper(a) → ok(a) 154.32/52.08
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.08
proper(b) → ok(b) 154.32/52.08
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.08
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.08
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.08
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.08
h(mark(z0)) → mark(h(z0)) 154.32/52.08
h(ok(z0)) → ok(h(z0)) 154.32/52.08
active(h(z0)) → mark(g(z0, z0)) 154.32/52.08
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.08
active(f(z0, z0)) → mark(h(a)) 154.32/52.08
active(a) → mark(b) 154.32/52.08
active(h(z0)) → h(active(z0)) 154.32/52.08
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.08
active(f(z0, z1)) → f(active(z0), z1)
And the Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.08
H(mark(z0)) → c7(H(z0)) 154.32/52.08
H(ok(z0)) → c8(H(z0)) 154.32/52.08
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.08
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.08
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.08
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.08
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.08
ACTIVE(g(a, z0)) → c1 154.32/52.08
ACTIVE(f(z0, z0)) → c2 154.32/52.08
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.08
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.08
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.08
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.08
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.08
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.08
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.08
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.08
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.08
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.08
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.08
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.08
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.08
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.08
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.08
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.08
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.08
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.08
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.08
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.08
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.08
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.08
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.08
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.08
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.08
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.08
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.08
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.08
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.08
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.08
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.08
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.08
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.08
TOP(mark(a)) → c18(TOP(ok(a))) 154.32/52.08
TOP(mark(b)) → c18(TOP(ok(b)))
The order we found is given by the following interpretation:
Polynomial interpretation : 154.32/52.08

POL(ACTIVE(x1)) = 0    154.32/52.08
POL(F(x1, x2)) = 0    154.32/52.08
POL(G(x1, x2)) = 0    154.32/52.08
POL(H(x1)) = 0    154.32/52.08
POL(PROPER(x1)) = 0    154.32/52.08
POL(TOP(x1)) = x1    154.32/52.08
POL(a) = [1]    154.32/52.08
POL(active(x1)) = 0    154.32/52.08
POL(b) = 0    154.32/52.08
POL(c(x1)) = x1    154.32/52.08
POL(c1) = 0    154.32/52.08
POL(c10(x1)) = x1    154.32/52.08
POL(c11(x1)) = x1    154.32/52.08
POL(c12(x1)) = x1    154.32/52.08
POL(c13(x1)) = x1    154.32/52.08
POL(c13(x1, x2)) = x1 + x2    154.32/52.08
POL(c14(x1, x2)) = x1 + x2    154.32/52.08
POL(c14(x1, x2, x3)) = x1 + x2 + x3    154.32/52.08
POL(c15(x1)) = x1    154.32/52.08
POL(c16(x1, x2)) = x1 + x2    154.32/52.08
POL(c16(x1, x2, x3)) = x1 + x2 + x3    154.32/52.08
POL(c18(x1)) = x1    154.32/52.08
POL(c18(x1, x2)) = x1 + x2    154.32/52.08
POL(c19(x1, x2)) = x1 + x2    154.32/52.08
POL(c2) = 0    154.32/52.08
POL(c3(x1)) = x1    154.32/52.08
POL(c4(x1)) = x1    154.32/52.08
POL(c4(x1, x2)) = x1 + x2    154.32/52.08
POL(c5(x1)) = x1    154.32/52.08
POL(c5(x1, x2)) = x1 + x2    154.32/52.08
POL(c6(x1)) = x1    154.32/52.08
POL(c6(x1, x2)) = x1 + x2    154.32/52.08
POL(c7(x1)) = x1    154.32/52.08
POL(c8(x1)) = x1    154.32/52.08
POL(c9(x1)) = x1    154.32/52.08
POL(f(x1, x2)) = 0    154.32/52.08
POL(g(x1, x2)) = 0    154.32/52.08
POL(h(x1)) = 0    154.32/52.08
POL(mark(x1)) = x1    154.32/52.08
POL(ok(x1)) = 0    154.32/52.08
POL(proper(x1)) = 0   
154.32/52.08
154.32/52.08

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.08
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.08
active(f(z0, z0)) → mark(h(a)) 154.32/52.08
active(a) → mark(b) 154.32/52.08
active(h(z0)) → h(active(z0)) 154.32/52.08
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.08
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.08
h(mark(z0)) → mark(h(z0)) 154.32/52.08
h(ok(z0)) → ok(h(z0)) 154.32/52.08
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.08
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.08
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.08
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.08
proper(h(z0)) → h(proper(z0)) 154.32/52.08
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.08
proper(a) → ok(a) 154.32/52.08
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.08
proper(b) → ok(b) 154.32/52.08
top(mark(z0)) → top(proper(z0)) 154.32/52.08
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.08
H(mark(z0)) → c7(H(z0)) 154.32/52.08
H(ok(z0)) → c8(H(z0)) 154.32/52.08
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.08
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.08
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.08
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.08
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.08
ACTIVE(g(a, z0)) → c1 154.32/52.08
ACTIVE(f(z0, z0)) → c2 154.32/52.08
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.08
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.08
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.08
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.08
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.08
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.08
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.08
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.08
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.08
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.08
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.08
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.08
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.08
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.08
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.08
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.08
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.08
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.08
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.08
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.08
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.08
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.08
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.08
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.08
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.08
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.08
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.08
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.08
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.08
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.08
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.08
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.08
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.08
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.08
TOP(mark(a)) → c18(TOP(ok(a))) 154.32/52.08
TOP(mark(b)) → c18(TOP(ok(b)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.08
H(mark(z0)) → c7(H(z0)) 154.32/52.08
H(ok(z0)) → c8(H(z0)) 154.32/52.08
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.08
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.08
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.08
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.08
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 154.32/52.08
ACTIVE(g(a, z0)) → c1 154.32/52.08
ACTIVE(f(z0, z0)) → c2 154.32/52.08
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.08
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.08
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.08
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.08
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.08
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.08
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.08
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.08
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.08
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.08
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.08
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.08
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.08
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.08
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.08
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.08
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.08
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.08
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.08
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.08
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.08
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.09
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.09
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.09
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.09
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.09
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.09
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.09
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.09
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.09
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.09
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.09
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.09
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.09
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.09
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.09
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
K tuples:

TOP(mark(b)) → c18(TOP(ok(b))) 154.32/52.09
TOP(mark(a)) → c18(TOP(ok(a)))
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c19, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13, c14, c14, c16, c16, c18, c18

154.32/52.09
154.32/52.09

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

Use narrowing to replace TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(h(z0))) → c19(TOP(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.09
TOP(ok(g(a, z0))) → c19(TOP(mark(f(b, z0))), ACTIVE(g(a, z0))) 154.32/52.09
TOP(ok(f(z0, z0))) → c19(TOP(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.09
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a)) 154.32/52.09
TOP(ok(h(z0))) → c19(TOP(h(active(z0))), ACTIVE(h(z0))) 154.32/52.09
TOP(ok(g(z0, z1))) → c19(TOP(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.09
TOP(ok(f(z0, z1))) → c19(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
154.32/52.09
154.32/52.09

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.09
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.09
active(f(z0, z0)) → mark(h(a)) 154.32/52.09
active(a) → mark(b) 154.32/52.09
active(h(z0)) → h(active(z0)) 154.32/52.09
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.09
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.09
h(mark(z0)) → mark(h(z0)) 154.32/52.09
h(ok(z0)) → ok(h(z0)) 154.32/52.09
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.09
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.09
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.09
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.09
proper(h(z0)) → h(proper(z0)) 154.32/52.09
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.09
proper(a) → ok(a) 154.32/52.09
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.09
proper(b) → ok(b) 154.32/52.09
top(mark(z0)) → top(proper(z0)) 154.32/52.09
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.09
H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.09
ACTIVE(g(a, z0)) → c1 154.32/52.09
ACTIVE(f(z0, z0)) → c2 154.32/52.09
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.09
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.09
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.09
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.09
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.09
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.09
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.09
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.09
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.09
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.09
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.09
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.09
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.09
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.09
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.09
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.09
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.09
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.09
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.09
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.09
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.09
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.09
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.09
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.09
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.09
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.09
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.09
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.09
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.09
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.09
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.09
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.09
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.09
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.09
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.09
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.09
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.09
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.09
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.09
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.09
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.09
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.09
TOP(mark(a)) → c18(TOP(ok(a))) 154.32/52.09
TOP(mark(b)) → c18(TOP(ok(b))) 154.32/52.09
TOP(ok(h(z0))) → c19(TOP(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.09
TOP(ok(g(a, z0))) → c19(TOP(mark(f(b, z0))), ACTIVE(g(a, z0))) 154.32/52.09
TOP(ok(f(z0, z0))) → c19(TOP(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.09
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a)) 154.32/52.09
TOP(ok(h(z0))) → c19(TOP(h(active(z0))), ACTIVE(h(z0))) 154.32/52.09
TOP(ok(g(z0, z1))) → c19(TOP(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.09
TOP(ok(f(z0, z1))) → c19(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
S tuples:

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.09
H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.09
ACTIVE(g(a, z0)) → c1 154.32/52.09
ACTIVE(f(z0, z0)) → c2 154.32/52.09
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.09
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.09
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.09
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.09
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.09
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.09
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.09
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.09
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.09
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.09
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.09
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.09
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.09
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.09
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.09
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.09
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.09
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.09
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.09
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.09
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.09
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.09
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.09
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.09
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.09
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.09
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.09
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.09
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.09
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.09
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.09
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.09
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.09
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.09
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.09
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.09
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.09
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.09
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.09
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.09
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.09
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.09
TOP(ok(h(z0))) → c19(TOP(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.09
TOP(ok(g(a, z0))) → c19(TOP(mark(f(b, z0))), ACTIVE(g(a, z0))) 154.32/52.09
TOP(ok(f(z0, z0))) → c19(TOP(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.09
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a)) 154.32/52.09
TOP(ok(h(z0))) → c19(TOP(h(active(z0))), ACTIVE(h(z0))) 154.32/52.09
TOP(ok(g(z0, z1))) → c19(TOP(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.09
TOP(ok(f(z0, z1))) → c19(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
K tuples:

TOP(mark(b)) → c18(TOP(ok(b))) 154.32/52.09
TOP(mark(a)) → c18(TOP(ok(a)))
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c1, c2, c4, c4, c5, c5, c3, c6, c6, c15, c13, c13, c14, c14, c16, c16, c18, c18, c19

154.32/52.09
154.32/52.09

(59) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(h(z0)) → c(G(z0, z0)) 154.32/52.09
ACTIVE(g(a, z0)) → c1 154.32/52.09
ACTIVE(f(z0, z0)) → c2 154.32/52.09
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.09
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.09
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1))) 154.32/52.09
ACTIVE(h(a)) → c4(H(mark(b))) 154.32/52.09
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.09
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.09
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.09
ACTIVE(g(a, x1)) → c5(G(mark(b), x1)) 154.32/52.09
ACTIVE(h(g(a, z0))) → c3(H(mark(f(b, z0)))) 154.32/52.09
ACTIVE(h(g(a, z0))) → c3(ACTIVE(g(a, z0))) 154.32/52.09
ACTIVE(g(g(a, z0), x1)) → c3(G(mark(f(b, z0)), x1)) 154.32/52.09
ACTIVE(g(g(a, z0), x1)) → c3(ACTIVE(g(a, z0))) 154.32/52.09
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0))) 154.32/52.09
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0))) 154.32/52.09
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1))) 154.32/52.09
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1))) 154.32/52.09
ACTIVE(f(a, x1)) → c6(F(mark(b), x1)) 154.32/52.09
ACTIVE(f(g(a, z0), x1)) → c15(F(mark(f(b, z0)), x1)) 154.32/52.09
ACTIVE(f(g(a, z0), x1)) → c15(ACTIVE(g(a, z0))) 154.32/52.09
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0))) 154.32/52.09
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.09
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.09
PROPER(h(a)) → c13(H(ok(a))) 154.32/52.09
PROPER(h(b)) → c13(H(ok(b))) 154.32/52.09
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.09
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.09
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.09
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.09
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0)) 154.32/52.09
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0)) 154.32/52.09
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1)) 154.32/52.09
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1)) 154.32/52.09
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0))) 154.32/52.09
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1))) 154.32/52.09
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1))) 154.32/52.09
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1)) 154.32/52.09
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1)) 154.32/52.09
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0)) 154.32/52.09
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0)) 154.32/52.09
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1)) 154.32/52.09
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1)) 154.32/52.09
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0))) 154.32/52.09
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1))) 154.32/52.09
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1))) 154.32/52.09
TOP(ok(h(z0))) → c19(TOP(mark(g(z0, z0))), ACTIVE(h(z0))) 154.32/52.09
TOP(ok(g(a, z0))) → c19(TOP(mark(f(b, z0))), ACTIVE(g(a, z0))) 154.32/52.09
TOP(ok(f(z0, z0))) → c19(TOP(mark(h(a))), ACTIVE(f(z0, z0))) 154.32/52.09
TOP(ok(h(z0))) → c19(TOP(h(active(z0))), ACTIVE(h(z0))) 154.32/52.09
TOP(ok(g(z0, z1))) → c19(TOP(g(active(z0), z1)), ACTIVE(g(z0, z1))) 154.32/52.09
TOP(ok(f(z0, z1))) → c19(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
154.32/52.09
154.32/52.09

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.09
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.09
active(f(z0, z0)) → mark(h(a)) 154.32/52.09
active(a) → mark(b) 154.32/52.09
active(h(z0)) → h(active(z0)) 154.32/52.09
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.09
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.09
h(mark(z0)) → mark(h(z0)) 154.32/52.09
h(ok(z0)) → ok(h(z0)) 154.32/52.09
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.09
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.09
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.09
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.09
proper(h(z0)) → h(proper(z0)) 154.32/52.09
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.09
proper(a) → ok(a) 154.32/52.09
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.09
proper(b) → ok(b) 154.32/52.09
top(mark(z0)) → top(proper(z0)) 154.32/52.09
top(ok(z0)) → top(active(z0))
Tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.09
TOP(mark(a)) → c18(TOP(ok(a))) 154.32/52.09
TOP(mark(b)) → c18(TOP(ok(b))) 154.32/52.09
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a))
S tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.09
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a))
K tuples:

TOP(mark(b)) → c18(TOP(ok(b))) 154.32/52.09
TOP(mark(a)) → c18(TOP(ok(a)))
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

H, G, F, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c18, c19

154.32/52.09
154.32/52.09

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

Removed 2 trailing tuple parts
154.32/52.09
154.32/52.09

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.09
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.09
active(f(z0, z0)) → mark(h(a)) 154.32/52.09
active(a) → mark(b) 154.32/52.09
active(h(z0)) → h(active(z0)) 154.32/52.09
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.09
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.09
h(mark(z0)) → mark(h(z0)) 154.32/52.09
h(ok(z0)) → ok(h(z0)) 154.32/52.09
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.09
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.09
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.09
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.09
proper(h(z0)) → h(proper(z0)) 154.32/52.09
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.09
proper(a) → ok(a) 154.32/52.09
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.09
proper(b) → ok(b) 154.32/52.09
top(mark(z0)) → top(proper(z0)) 154.32/52.09
top(ok(z0)) → top(active(z0))
Tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.09
TOP(mark(a)) → c18(TOP(ok(a))) 154.32/52.09
TOP(mark(b)) → c18 154.32/52.09
TOP(ok(a)) → c19(TOP(mark(b)))
S tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.09
TOP(ok(a)) → c19(TOP(mark(b)))
K tuples:

TOP(mark(a)) → c18(TOP(ok(a))) 154.32/52.09
TOP(mark(b)) → c18
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

H, G, F, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c18, c18, c19

154.32/52.09
154.32/52.09

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

Removed 3 trailing nodes:

TOP(ok(a)) → c19(TOP(mark(b))) 154.32/52.09
TOP(mark(a)) → c18(TOP(ok(a))) 154.32/52.09
TOP(mark(b)) → c18
154.32/52.09
154.32/52.09

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.09
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.09
active(f(z0, z0)) → mark(h(a)) 154.32/52.09
active(a) → mark(b) 154.32/52.09
active(h(z0)) → h(active(z0)) 154.32/52.09
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.09
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.09
h(mark(z0)) → mark(h(z0)) 154.32/52.09
h(ok(z0)) → ok(h(z0)) 154.32/52.09
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.09
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.09
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.09
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.09
proper(h(z0)) → h(proper(z0)) 154.32/52.09
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.09
proper(a) → ok(a) 154.32/52.09
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.09
proper(b) → ok(b) 154.32/52.09
top(mark(z0)) → top(proper(z0)) 154.32/52.09
top(ok(z0)) → top(active(z0))
Tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1))
S tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1))
K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

H, G, F

Compound Symbols:

c7, c8, c9, c10, c11, c12

154.32/52.09
154.32/52.09

(65) 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.

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 154.32/52.09

POL(F(x1, x2)) = x1    154.32/52.09
POL(G(x1, x2)) = 0    154.32/52.09
POL(H(x1)) = [2]x1    154.32/52.09
POL(c10(x1)) = x1    154.32/52.09
POL(c11(x1)) = x1    154.32/52.09
POL(c12(x1)) = x1    154.32/52.09
POL(c7(x1)) = x1    154.32/52.09
POL(c8(x1)) = x1    154.32/52.09
POL(c9(x1)) = x1    154.32/52.09
POL(mark(x1)) = [1] + x1    154.32/52.09
POL(ok(x1)) = [1] + x1   
154.32/52.09
154.32/52.09

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.09
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.09
active(f(z0, z0)) → mark(h(a)) 154.32/52.09
active(a) → mark(b) 154.32/52.09
active(h(z0)) → h(active(z0)) 154.32/52.09
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.09
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.09
h(mark(z0)) → mark(h(z0)) 154.32/52.09
h(ok(z0)) → ok(h(z0)) 154.32/52.09
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.09
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.09
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.09
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.09
proper(h(z0)) → h(proper(z0)) 154.32/52.09
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.09
proper(a) → ok(a) 154.32/52.09
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.09
proper(b) → ok(b) 154.32/52.09
top(mark(z0)) → top(proper(z0)) 154.32/52.09
top(ok(z0)) → top(active(z0))
Tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1))
S tuples:

G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1))
K tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1))
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

H, G, F

Compound Symbols:

c7, c8, c9, c10, c11, c12

154.32/52.09
154.32/52.09

(67) 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.

G(ok(z0), ok(z1)) → c10(G(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 154.32/52.09

POL(F(x1, x2)) = x2    154.32/52.09
POL(G(x1, x2)) = [2]x22    154.32/52.09
POL(H(x1)) = 0    154.32/52.09
POL(c10(x1)) = x1    154.32/52.09
POL(c11(x1)) = x1    154.32/52.09
POL(c12(x1)) = x1    154.32/52.09
POL(c7(x1)) = x1    154.32/52.09
POL(c8(x1)) = x1    154.32/52.09
POL(c9(x1)) = x1    154.32/52.09
POL(mark(x1)) = 0    154.32/52.09
POL(ok(x1)) = [1] + x1   
154.32/52.09
154.32/52.09

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.09
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.09
active(f(z0, z0)) → mark(h(a)) 154.32/52.09
active(a) → mark(b) 154.32/52.09
active(h(z0)) → h(active(z0)) 154.32/52.09
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.09
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.09
h(mark(z0)) → mark(h(z0)) 154.32/52.09
h(ok(z0)) → ok(h(z0)) 154.32/52.09
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.09
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.09
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.09
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.09
proper(h(z0)) → h(proper(z0)) 154.32/52.09
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.09
proper(a) → ok(a) 154.32/52.09
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.09
proper(b) → ok(b) 154.32/52.09
top(mark(z0)) → top(proper(z0)) 154.32/52.09
top(ok(z0)) → top(active(z0))
Tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1))
S tuples:

G(mark(z0), z1) → c9(G(z0, z1))
K tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1))
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

H, G, F

Compound Symbols:

c7, c8, c9, c10, c11, c12

154.32/52.09
154.32/52.09

(69) 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.

G(mark(z0), z1) → c9(G(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 154.32/52.09

POL(F(x1, x2)) = x1 + x2 + [3]x22 + [3]x1·x2 + [3]x12    154.32/52.09
POL(G(x1, x2)) = x2 + [3]x22 + [3]x1·x2 + x12    154.32/52.09
POL(H(x1)) = [3]x1 + [3]x12    154.32/52.09
POL(c10(x1)) = x1    154.32/52.09
POL(c11(x1)) = x1    154.32/52.09
POL(c12(x1)) = x1    154.32/52.09
POL(c7(x1)) = x1    154.32/52.09
POL(c8(x1)) = x1    154.32/52.09
POL(c9(x1)) = x1    154.32/52.09
POL(mark(x1)) = [1] + x1    154.32/52.09
POL(ok(x1)) = x1   
154.32/52.09
154.32/52.09

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0)) 154.32/52.09
active(g(a, z0)) → mark(f(b, z0)) 154.32/52.09
active(f(z0, z0)) → mark(h(a)) 154.32/52.09
active(a) → mark(b) 154.32/52.09
active(h(z0)) → h(active(z0)) 154.32/52.09
active(g(z0, z1)) → g(active(z0), z1) 154.32/52.09
active(f(z0, z1)) → f(active(z0), z1) 154.32/52.09
h(mark(z0)) → mark(h(z0)) 154.32/52.09
h(ok(z0)) → ok(h(z0)) 154.32/52.09
g(mark(z0), z1) → mark(g(z0, z1)) 154.32/52.09
g(ok(z0), ok(z1)) → ok(g(z0, z1)) 154.32/52.09
f(mark(z0), z1) → mark(f(z0, z1)) 154.32/52.09
f(ok(z0), ok(z1)) → ok(f(z0, z1)) 154.32/52.09
proper(h(z0)) → h(proper(z0)) 154.32/52.09
proper(g(z0, z1)) → g(proper(z0), proper(z1)) 154.32/52.09
proper(a) → ok(a) 154.32/52.09
proper(f(z0, z1)) → f(proper(z0), proper(z1)) 154.32/52.09
proper(b) → ok(b) 154.32/52.09
top(mark(z0)) → top(proper(z0)) 154.32/52.09
top(ok(z0)) → top(active(z0))
Tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1))
S tuples:none
K tuples:

H(mark(z0)) → c7(H(z0)) 154.32/52.09
H(ok(z0)) → c8(H(z0)) 154.32/52.09
F(mark(z0), z1) → c11(F(z0, z1)) 154.32/52.09
F(ok(z0), ok(z1)) → c12(F(z0, z1)) 154.32/52.09
G(ok(z0), ok(z1)) → c10(G(z0, z1)) 154.32/52.09
G(mark(z0), z1) → c9(G(z0, z1))
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

H, G, F

Compound Symbols:

c7, c8, c9, c10, c11, c12

154.32/52.09
154.32/52.09

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

The set S is empty
154.32/52.09
154.32/52.09

(72) BOUNDS(O(1), O(1))

154.32/52.09
154.32/52.09
154.70/52.18 EOF