70.77/24.52 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml

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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS

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

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↳2 CdtProblem

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

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↳4 CdtProblem

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

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↳6 CdtProblem

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

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↳8 CdtProblem

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

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↳10 CdtProblem

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

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↳12 CdtProblem

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

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↳14 CdtProblem

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

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↳16 CdtProblem

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↳17 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳18 CdtProblem

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

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↳20 CdtProblem

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

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↳22 CdtProblem

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↳23 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳24 CdtProblem

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

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↳26 CdtProblem

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

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↳28 CdtProblem

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

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↳30 CdtProblem

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↳31 CdtUnreachableProof (⇔)

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↳32 CdtProblem

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↳33 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳34 CdtProblem

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↳35 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳36 CdtProblem

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

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↳38 CdtProblem

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

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↳40 CdtProblem

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

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↳42 BOUNDS(O(1), O(1))

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(0) Obligation:

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

active(f(a, X, X)) → mark(f(X, b, b)) 70.77/24.52
active(b) → mark(a) 70.77/24.52
active(f(X1, X2, X3)) → f(X1, active(X2), X3) 70.77/24.52
f(X1, mark(X2), X3) → mark(f(X1, X2, X3)) 70.77/24.52
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3)) 70.77/24.52
proper(a) → ok(a) 70.77/24.52
proper(b) → ok(b) 70.77/24.52
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3)) 70.77/24.52
top(mark(X)) → top(proper(X)) 70.77/24.55
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

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(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.55
active(b) → mark(a) 70.77/24.55
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.55
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.55
proper(a) → ok(a) 70.77/24.55
proper(b) → ok(b) 70.77/24.55
top(mark(z0)) → top(proper(z0)) 70.77/24.55
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(a, z0, z0)) → c(F(z0, b, b)) 70.77/24.55
ACTIVE(f(z0, z1, z2)) → c2(F(z0, active(z1), z2), ACTIVE(z1)) 70.77/24.55
F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.55
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.55
PROPER(f(z0, z1, z2)) → c5(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 70.77/24.55
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.55
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(a, z0, z0)) → c(F(z0, b, b)) 70.77/24.55
ACTIVE(f(z0, z1, z2)) → c2(F(z0, active(z1), z2), ACTIVE(z1)) 70.77/24.55
F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.55
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.55
PROPER(f(z0, z1, z2)) → c5(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 70.77/24.55
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.55
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing tuple parts

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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.55
active(b) → mark(a) 70.77/24.55
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.55
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.55
proper(a) → ok(a) 70.77/24.55
proper(b) → ok(b) 70.77/24.55
top(mark(z0)) → top(proper(z0)) 70.77/24.55
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0, z1, z2)) → c2(F(z0, active(z1), z2), ACTIVE(z1)) 70.77/24.55
F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.55
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.55
PROPER(f(z0, z1, z2)) → c5(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 70.77/24.55
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.55
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.55
ACTIVE(f(a, z0, z0)) → c

S tuples:

ACTIVE(f(z0, z1, z2)) → c2(F(z0, active(z1), z2), ACTIVE(z1)) 70.77/24.55
F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.55
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.55
PROPER(f(z0, z1, z2)) → c5(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 70.77/24.55
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.55
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.55
ACTIVE(f(a, z0, z0)) → c

K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

ACTIVE(f(a, z0, z0)) → c

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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.55
active(b) → mark(a) 70.77/24.55
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.55
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.55
proper(a) → ok(a) 70.77/24.55
proper(b) → ok(b) 70.77/24.55
top(mark(z0)) → top(proper(z0)) 70.77/24.55
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0, z1, z2)) → c2(F(z0, active(z1), z2), ACTIVE(z1)) 70.77/24.55
F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.55
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.55
PROPER(f(z0, z1, z2)) → c5(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 70.77/24.55
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.55
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.55
ACTIVE(f(a, z0, z0)) → c

S tuples:

ACTIVE(f(z0, z1, z2)) → c2(F(z0, active(z1), z2), ACTIVE(z1)) 70.77/24.55
F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.55
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.55
PROPER(f(z0, z1, z2)) → c5(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 70.77/24.55
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.55
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.55
ACTIVE(f(a, z0, z0)) → c

K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(f(z0, z1, z2)) → c2(F(z0, active(z1), z2), ACTIVE(z1)) by

ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.55
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2), ACTIVE(b)) 70.77/24.55
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2)))

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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.55
active(b) → mark(a) 70.77/24.55
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.55
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.55
proper(a) → ok(a) 70.77/24.55
proper(b) → ok(b) 70.77/24.55
top(mark(z0)) → top(proper(z0)) 70.77/24.55
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.55
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.55
PROPER(f(z0, z1, z2)) → c5(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 70.77/24.55
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.55
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.55
ACTIVE(f(a, z0, z0)) → c 70.77/24.55
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.55
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2), ACTIVE(b)) 70.77/24.55
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2)))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.55
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.55
PROPER(f(z0, z1, z2)) → c5(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 70.77/24.55
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.55
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.55
ACTIVE(f(a, z0, z0)) → c 70.77/24.55
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.55
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2), ACTIVE(b)) 70.77/24.55
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2)))

K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

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

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

Removed 1 trailing tuple parts

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(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.55
active(b) → mark(a) 70.77/24.55
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.55
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.55
proper(a) → ok(a) 70.77/24.55
proper(b) → ok(b) 70.77/24.55
top(mark(z0)) → top(proper(z0)) 70.77/24.55
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.55
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.55
PROPER(f(z0, z1, z2)) → c5(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 70.77/24.55
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.55
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.55
ACTIVE(f(a, z0, z0)) → c 70.77/24.55
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.55
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.55
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.55
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.55
PROPER(f(z0, z1, z2)) → c5(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 70.77/24.55
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.55
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.55
ACTIVE(f(a, z0, z0)) → c 70.77/24.55
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.55
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.55
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2))

K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

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

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

Removed 1 trailing nodes:

ACTIVE(f(a, z0, z0)) → c

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(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.55
active(b) → mark(a) 70.77/24.55
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.55
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.55
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.55
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.55
proper(a) → ok(a) 70.77/24.55
proper(b) → ok(b) 70.77/24.55
top(mark(z0)) → top(proper(z0)) 70.77/24.55
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.55
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.55
PROPER(f(z0, z1, z2)) → c5(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 70.77/24.55
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.55
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.55
ACTIVE(f(a, z0, z0)) → c 70.77/24.55
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.55
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.55
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.55
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.55
PROPER(f(z0, z1, z2)) → c5(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 70.77/24.55
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.55
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.55
ACTIVE(f(a, z0, z0)) → c 70.77/24.55
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.55
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2))

K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

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

70.77/24.56

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

Use narrowing to replace PROPER(f(z0, z1, z2)) → c5(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2))

70.77/24.56

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

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

70.77/24.56

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

Removed 6 trailing tuple parts

70.77/24.56

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

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

70.77/24.56

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

Removed 1 trailing nodes:

ACTIVE(f(a, z0, z0)) → c

70.77/24.56

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

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

70.77/24.56

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

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

TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a)) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b)), PROPER(b))

70.77/24.56

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a)) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b)), PROPER(b))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a)) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b)), PROPER(b))

K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

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

70.77/24.56

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

Removed 2 trailing tuple parts

70.77/24.56

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a))) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b)))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a))) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b)))

K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

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

70.77/24.56

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

Removed 1 trailing nodes:

ACTIVE(f(a, z0, z0)) → c

70.77/24.56

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a))) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b)))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a))) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b)))

K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

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

70.77/24.56

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

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

TOP(mark(b)) → c8(TOP(ok(b)))

We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2)

And the Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a))) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b)))

The order we found is given by the following interpretation:
Polynomial interpretation : 70.77/24.56

POL(ACTIVE(x₁)) = 0 70.77/24.56
POL(F(x₁, x₂, x₃)) = 0 70.77/24.56
POL(PROPER(x₁)) = 0 70.77/24.56
POL(TOP(x₁)) = [4]x₁ 70.77/24.56
POL(a) = 0 70.77/24.56
POL(active(x₁)) = 0 70.77/24.56
POL(b) = [4] 70.77/24.56
POL(c) = 0 70.77/24.56
POL(c2(x₁)) = x₁ 70.77/24.56
POL(c2(x₁, x₂)) = x₁ + x₂ 70.77/24.56
POL(c3(x₁)) = x₁ 70.77/24.56
POL(c4(x₁)) = x₁ 70.77/24.56
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 70.77/24.56
POL(c5(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄ 70.77/24.56
POL(c8(x₁)) = x₁ 70.77/24.56
POL(c8(x₁, x₂)) = x₁ + x₂ 70.77/24.56
POL(c9(x₁, x₂)) = x₁ + x₂ 70.77/24.56
POL(f(x₁, x₂, x₃)) = 0 70.77/24.56
POL(mark(x₁)) = x₁ 70.77/24.56
POL(ok(x₁)) = 0 70.77/24.56
POL(proper(x₁)) = 0

70.77/24.56

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a))) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b)))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a)))

K tuples:

TOP(mark(b)) → c8(TOP(ok(b)))

Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

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

70.77/24.56

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

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

TOP(mark(a)) → c8(TOP(ok(a)))

We considered the (Usable) Rules:

proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2)

And the Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a))) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b)))

The order we found is given by the following interpretation:
Polynomial interpretation : 70.77/24.56

POL(ACTIVE(x₁)) = 0 70.77/24.56
POL(F(x₁, x₂, x₃)) = 0 70.77/24.56
POL(PROPER(x₁)) = 0 70.77/24.56
POL(TOP(x₁)) = [2]x₁ 70.77/24.56
POL(a) = 0 70.77/24.56
POL(active(x₁)) = x₁ 70.77/24.56
POL(b) = [1] 70.77/24.56
POL(c) = 0 70.77/24.56
POL(c2(x₁)) = x₁ 70.77/24.56
POL(c2(x₁, x₂)) = x₁ + x₂ 70.77/24.56
POL(c3(x₁)) = x₁ 70.77/24.56
POL(c4(x₁)) = x₁ 70.77/24.56
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 70.77/24.56
POL(c5(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄ 70.77/24.56
POL(c8(x₁)) = x₁ 70.77/24.56
POL(c8(x₁, x₂)) = x₁ + x₂ 70.77/24.56
POL(c9(x₁, x₂)) = x₁ + x₂ 70.77/24.56
POL(f(x₁, x₂, x₃)) = [1] 70.77/24.56
POL(mark(x₁)) = [1] 70.77/24.56
POL(ok(x₁)) = x₁ 70.77/24.56
POL(proper(x₁)) = 0

70.77/24.56

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a))) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b)))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))

K tuples:

TOP(mark(b)) → c8(TOP(ok(b))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a)))

Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

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

70.77/24.56

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

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

TOP(ok(f(a, z0, z0))) → c9(TOP(mark(f(z0, b, b))), ACTIVE(f(a, z0, z0))) 70.77/24.56
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b)) 70.77/24.56
TOP(ok(f(z0, z1, z2))) → c9(TOP(f(z0, active(z1), z2)), ACTIVE(f(z0, z1, z2)))

70.77/24.56

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a))) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b))) 70.77/24.56
TOP(ok(f(a, z0, z0))) → c9(TOP(mark(f(z0, b, b))), ACTIVE(f(a, z0, z0))) 70.77/24.56
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b)) 70.77/24.56
TOP(ok(f(z0, z1, z2))) → c9(TOP(f(z0, active(z1), z2)), ACTIVE(f(z0, z1, z2)))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(ok(f(a, z0, z0))) → c9(TOP(mark(f(z0, b, b))), ACTIVE(f(a, z0, z0))) 70.77/24.56
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b)) 70.77/24.56
TOP(ok(f(z0, z1, z2))) → c9(TOP(f(z0, active(z1), z2)), ACTIVE(f(z0, z1, z2)))

K tuples:

TOP(mark(b)) → c8(TOP(ok(b))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a)))

Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, ACTIVE, PROPER, TOP

Compound Symbols:

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

70.77/24.56

(31) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(f(a, z0, z0)) → c 70.77/24.56
ACTIVE(f(x0, f(a, z0, z0), x2)) → c2(F(x0, mark(f(z0, b, b)), x2), ACTIVE(f(a, z0, z0))) 70.77/24.56
ACTIVE(f(x0, f(z0, z1, z2), x2)) → c2(F(x0, f(z0, active(z1), z2), x2), ACTIVE(f(z0, z1, z2))) 70.77/24.56
ACTIVE(f(x0, b, x2)) → c2(F(x0, mark(a), x2)) 70.77/24.56
PROPER(f(x0, x1, f(z0, z1, z2))) → c5(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 70.77/24.56
PROPER(f(x0, f(z0, z1, z2), x2)) → c5(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 70.77/24.56
PROPER(f(f(z0, z1, z2), x1, x2)) → c5(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(x0, x1, a)) → c5(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, x1, b)) → c5(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 70.77/24.56
PROPER(f(x0, a, x2)) → c5(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(x0, b, x2)) → c5(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 70.77/24.56
PROPER(f(a, x1, x2)) → c5(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
PROPER(f(b, x1, x2)) → c5(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 70.77/24.56
TOP(mark(f(z0, z1, z2))) → c8(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 70.77/24.56
TOP(ok(f(a, z0, z0))) → c9(TOP(mark(f(z0, b, b))), ACTIVE(f(a, z0, z0))) 70.77/24.56
TOP(ok(f(z0, z1, z2))) → c9(TOP(f(z0, active(z1), z2)), ACTIVE(f(z0, z1, z2)))

70.77/24.56

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a))) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b))) 70.77/24.56
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))

K tuples:

TOP(mark(b)) → c8(TOP(ok(b))) 70.77/24.56
TOP(mark(a)) → c8(TOP(ok(a)))

Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP

Compound Symbols:

c3, c4, c8, c9

70.77/24.56

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

Removed 2 trailing tuple parts

70.77/24.56

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b))) 70.77/24.56
TOP(mark(a)) → c8 70.77/24.56
TOP(ok(b)) → c9(TOP(mark(a)))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2)) 70.77/24.56
TOP(ok(b)) → c9(TOP(mark(a)))

K tuples:

TOP(mark(b)) → c8(TOP(ok(b))) 70.77/24.56
TOP(mark(a)) → c8

Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP

Compound Symbols:

c3, c4, c8, c8, c9

70.77/24.56

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

Removed 3 trailing nodes:

TOP(mark(a)) → c8 70.77/24.56
TOP(ok(b)) → c9(TOP(mark(a))) 70.77/24.56
TOP(mark(b)) → c8(TOP(ok(b)))

70.77/24.56

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2))

S tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

70.77/24.56

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

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

F(z0, mark(z1), z2) → c3(F(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2))

The order we found is given by the following interpretation:
Polynomial interpretation : 70.77/24.56

POL(F(x₁, x₂, x₃)) = [4]x₂ 70.77/24.56
POL(c3(x₁)) = x₁ 70.77/24.56
POL(c4(x₁)) = x₁ 70.77/24.56
POL(mark(x₁)) = [4] + x₁ 70.77/24.56
POL(ok(x₁)) = x₁

70.77/24.56

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2))

S tuples:

F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2))

K tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2))

Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

70.77/24.56

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

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

F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2))

The order we found is given by the following interpretation:
Polynomial interpretation : 70.77/24.56

POL(F(x₁, x₂, x₃)) = x₃ 70.77/24.56
POL(c3(x₁)) = x₁ 70.77/24.56
POL(c4(x₁)) = x₁ 70.77/24.56
POL(mark(x₁)) = x₁ 70.77/24.56
POL(ok(x₁)) = [1] + x₁

70.77/24.56

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(a, z0, z0)) → mark(f(z0, b, b)) 70.77/24.56
active(b) → mark(a) 70.77/24.56
active(f(z0, z1, z2)) → f(z0, active(z1), z2) 70.77/24.56
f(z0, mark(z1), z2) → mark(f(z0, z1, z2)) 70.77/24.56
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 70.77/24.56
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 70.77/24.56
proper(a) → ok(a) 70.77/24.56
proper(b) → ok(b) 70.77/24.56
top(mark(z0)) → top(proper(z0)) 70.77/24.56
top(ok(z0)) → top(active(z0))

Tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2))

S tuples:none
K tuples:

F(z0, mark(z1), z2) → c3(F(z0, z1, z2)) 70.77/24.56
F(ok(z0), ok(z1), ok(z2)) → c4(F(z0, z1, z2))

Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

70.77/24.56

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

The set S is empty

70.77/24.56

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

70.77/24.56