(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(x1) → x1
a(b(b(x1))) → b(b(b(c(a(x1)))))
b(c(x1)) → a(x1)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(b(z0))) → b(b(b(c(a(z0)))))
b(c(z0)) → a(z0)
Tuples:
A(b(b(z0))) → c2(B(b(b(c(a(z0))))), B(b(c(a(z0)))), B(c(a(z0))), A(z0))
B(c(z0)) → c3(A(z0))
S tuples:
A(b(b(z0))) → c2(B(b(b(c(a(z0))))), B(b(c(a(z0)))), B(c(a(z0))), A(z0))
B(c(z0)) → c3(A(z0))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c2, c3
(3) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
A(
b(
b(
z0))) →
c2(
B(
b(
b(
c(
a(
z0))))),
B(
b(
c(
a(
z0)))),
B(
c(
a(
z0))),
A(
z0)) by
A(b(b(x0))) → c2(B(b(a(a(x0)))), B(b(c(a(x0)))), B(c(a(x0))), A(x0))
A(b(b(z0))) → c2(B(b(b(c(z0)))), B(b(c(a(z0)))), B(c(a(z0))), A(z0))
A(b(b(b(b(z0))))) → c2(B(b(b(c(b(b(b(c(a(z0))))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(b(z0))) → b(b(b(c(a(z0)))))
b(c(z0)) → a(z0)
Tuples:
B(c(z0)) → c3(A(z0))
A(b(b(x0))) → c2(B(b(a(a(x0)))), B(b(c(a(x0)))), B(c(a(x0))), A(x0))
A(b(b(z0))) → c2(B(b(b(c(z0)))), B(b(c(a(z0)))), B(c(a(z0))), A(z0))
A(b(b(b(b(z0))))) → c2(B(b(b(c(b(b(b(c(a(z0))))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
S tuples:
B(c(z0)) → c3(A(z0))
A(b(b(x0))) → c2(B(b(a(a(x0)))), B(b(c(a(x0)))), B(c(a(x0))), A(x0))
A(b(b(z0))) → c2(B(b(b(c(z0)))), B(b(c(a(z0)))), B(c(a(z0))), A(z0))
A(b(b(b(b(z0))))) → c2(B(b(b(c(b(b(b(c(a(z0))))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
B, A
Compound Symbols:
c3, c2
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
A(
b(
b(
x0))) →
c2(
B(
b(
a(
a(
x0)))),
B(
b(
c(
a(
x0)))),
B(
c(
a(
x0))),
A(
x0)) by
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(a(x0)))), B(c(a(x0))), A(x0))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(a(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(b(z0))) → b(b(b(c(a(z0)))))
b(c(z0)) → a(z0)
Tuples:
B(c(z0)) → c3(A(z0))
A(b(b(z0))) → c2(B(b(b(c(z0)))), B(b(c(a(z0)))), B(c(a(z0))), A(z0))
A(b(b(b(b(z0))))) → c2(B(b(b(c(b(b(b(c(a(z0))))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(a(x0)))), B(c(a(x0))), A(x0))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(a(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
S tuples:
B(c(z0)) → c3(A(z0))
A(b(b(z0))) → c2(B(b(b(c(z0)))), B(b(c(a(z0)))), B(c(a(z0))), A(z0))
A(b(b(b(b(z0))))) → c2(B(b(b(c(b(b(b(c(a(z0))))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(a(x0)))), B(c(a(x0))), A(x0))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(a(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
B, A
Compound Symbols:
c3, c2
(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
A(
b(
b(
z0))) →
c2(
B(
b(
b(
c(
z0)))),
B(
b(
c(
a(
z0)))),
B(
c(
a(
z0))),
A(
z0)) by
A(b(b(z0))) → c2(B(b(a(z0))), B(b(c(a(z0)))), B(c(a(z0))), A(z0))
A(b(b(x0))) → c2(B(b(c(a(x0)))), B(c(a(x0))))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(b(z0))) → b(b(b(c(a(z0)))))
b(c(z0)) → a(z0)
Tuples:
B(c(z0)) → c3(A(z0))
A(b(b(b(b(z0))))) → c2(B(b(b(c(b(b(b(c(a(z0))))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(a(x0)))), B(c(a(x0))), A(x0))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(a(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(x0))) → c2(B(b(c(a(x0)))), B(c(a(x0))))
S tuples:
B(c(z0)) → c3(A(z0))
A(b(b(b(b(z0))))) → c2(B(b(b(c(b(b(b(c(a(z0))))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(a(x0)))), B(c(a(x0))), A(x0))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(a(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(x0))) → c2(B(b(c(a(x0)))), B(c(a(x0))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
B, A
Compound Symbols:
c3, c2, c2
(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
A(
b(
b(
b(
b(
z0))))) →
c2(
B(
b(
b(
c(
b(
b(
b(
c(
a(
z0))))))))),
B(
b(
c(
a(
b(
b(
z0)))))),
B(
c(
a(
b(
b(
z0))))),
A(
b(
b(
z0)))) by
A(b(b(b(b(x0))))) → c2(B(b(a(b(b(b(c(a(x0)))))))), B(b(c(a(b(b(x0)))))), B(c(a(b(b(x0))))), A(b(b(x0))))
A(b(b(b(b(x0))))) → c2(B(b(b(c(b(b(a(a(x0)))))))), B(b(c(a(b(b(x0)))))), B(c(a(b(b(x0))))), A(b(b(x0))))
A(b(b(b(b(z0))))) → c2(B(b(b(c(b(b(b(c(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(b(b(b(b(z0))))))) → c2(B(b(b(c(b(b(b(c(b(b(b(c(a(z0))))))))))))), B(b(c(a(b(b(b(b(z0)))))))), B(c(a(b(b(b(b(z0))))))), A(b(b(b(b(z0))))))
A(b(b(b(b(x0))))) → c2(B(b(c(a(b(b(x0)))))), B(c(a(b(b(x0))))))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(b(z0))) → b(b(b(c(a(z0)))))
b(c(z0)) → a(z0)
Tuples:
B(c(z0)) → c3(A(z0))
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(a(x0)))), B(c(a(x0))), A(x0))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(a(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(x0))) → c2(B(b(c(a(x0)))), B(c(a(x0))))
A(b(b(b(b(x0))))) → c2(B(b(b(c(b(b(a(a(x0)))))))), B(b(c(a(b(b(x0)))))), B(c(a(b(b(x0))))), A(b(b(x0))))
A(b(b(b(b(z0))))) → c2(B(b(b(c(b(b(b(c(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(b(b(b(b(z0))))))) → c2(B(b(b(c(b(b(b(c(b(b(b(c(a(z0))))))))))))), B(b(c(a(b(b(b(b(z0)))))))), B(c(a(b(b(b(b(z0))))))), A(b(b(b(b(z0))))))
A(b(b(b(b(x0))))) → c2(B(b(c(a(b(b(x0)))))), B(c(a(b(b(x0))))))
S tuples:
B(c(z0)) → c3(A(z0))
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(a(x0)))), B(c(a(x0))), A(x0))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(a(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(x0))) → c2(B(b(c(a(x0)))), B(c(a(x0))))
A(b(b(b(b(x0))))) → c2(B(b(b(c(b(b(a(a(x0)))))))), B(b(c(a(b(b(x0)))))), B(c(a(b(b(x0))))), A(b(b(x0))))
A(b(b(b(b(z0))))) → c2(B(b(b(c(b(b(b(c(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(b(b(b(b(z0))))))) → c2(B(b(b(c(b(b(b(c(b(b(b(c(a(z0))))))))))))), B(b(c(a(b(b(b(b(z0)))))))), B(c(a(b(b(b(b(z0))))))), A(b(b(b(b(z0))))))
A(b(b(b(b(x0))))) → c2(B(b(c(a(b(b(x0)))))), B(c(a(b(b(x0))))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
B, A
Compound Symbols:
c3, c2, c2
(11) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
B(
c(
z0)) →
c3(
A(
z0)) by
B(c(b(b(y0)))) → c3(A(b(b(y0))))
B(c(b(b(b(b(y0)))))) → c3(A(b(b(b(b(y0))))))
B(c(b(b(b(b(b(b(y0)))))))) → c3(A(b(b(b(b(b(b(y0))))))))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(b(z0))) → b(b(b(c(a(z0)))))
b(c(z0)) → a(z0)
Tuples:
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(a(x0)))), B(c(a(x0))), A(x0))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(a(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(x0))) → c2(B(b(c(a(x0)))), B(c(a(x0))))
A(b(b(b(b(x0))))) → c2(B(b(b(c(b(b(a(a(x0)))))))), B(b(c(a(b(b(x0)))))), B(c(a(b(b(x0))))), A(b(b(x0))))
A(b(b(b(b(z0))))) → c2(B(b(b(c(b(b(b(c(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(b(b(b(b(z0))))))) → c2(B(b(b(c(b(b(b(c(b(b(b(c(a(z0))))))))))))), B(b(c(a(b(b(b(b(z0)))))))), B(c(a(b(b(b(b(z0))))))), A(b(b(b(b(z0))))))
A(b(b(b(b(x0))))) → c2(B(b(c(a(b(b(x0)))))), B(c(a(b(b(x0))))))
B(c(b(b(y0)))) → c3(A(b(b(y0))))
B(c(b(b(b(b(y0)))))) → c3(A(b(b(b(b(y0))))))
B(c(b(b(b(b(b(b(y0)))))))) → c3(A(b(b(b(b(b(b(y0))))))))
S tuples:
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(a(x0)))), B(c(a(x0))), A(x0))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(a(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(x0))) → c2(B(b(c(a(x0)))), B(c(a(x0))))
A(b(b(b(b(x0))))) → c2(B(b(b(c(b(b(a(a(x0)))))))), B(b(c(a(b(b(x0)))))), B(c(a(b(b(x0))))), A(b(b(x0))))
A(b(b(b(b(z0))))) → c2(B(b(b(c(b(b(b(c(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(b(b(b(b(z0))))))) → c2(B(b(b(c(b(b(b(c(b(b(b(c(a(z0))))))))))))), B(b(c(a(b(b(b(b(z0)))))))), B(c(a(b(b(b(b(z0))))))), A(b(b(b(b(z0))))))
A(b(b(b(b(x0))))) → c2(B(b(c(a(b(b(x0)))))), B(c(a(b(b(x0))))))
B(c(b(b(y0)))) → c3(A(b(b(y0))))
B(c(b(b(b(b(y0)))))) → c3(A(b(b(b(b(y0))))))
B(c(b(b(b(b(b(b(y0)))))))) → c3(A(b(b(b(b(b(b(y0))))))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c2, c2, c3
(13) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(a(x0)))), B(c(a(x0))), A(x0))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(a(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(x0))) → c2(B(b(c(a(x0)))), B(c(a(x0))))
A(b(b(b(b(x0))))) → c2(B(b(b(c(b(b(a(a(x0)))))))), B(b(c(a(b(b(x0)))))), B(c(a(b(b(x0))))), A(b(b(x0))))
A(b(b(b(b(z0))))) → c2(B(b(b(c(b(b(b(c(z0)))))))), B(b(c(a(b(b(z0)))))), B(c(a(b(b(z0))))), A(b(b(z0))))
A(b(b(b(b(b(b(z0))))))) → c2(B(b(b(c(b(b(b(c(b(b(b(c(a(z0))))))))))))), B(b(c(a(b(b(b(b(z0)))))))), B(c(a(b(b(b(b(z0))))))), A(b(b(b(b(z0))))))
A(b(b(b(b(x0))))) → c2(B(b(c(a(b(b(x0)))))), B(c(a(b(b(x0))))))
B(c(b(b(y0)))) → c3(A(b(b(y0))))
B(c(b(b(b(b(y0)))))) → c3(A(b(b(b(b(y0))))))
B(c(b(b(b(b(b(b(y0)))))))) → c3(A(b(b(b(b(b(b(y0))))))))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(b(z0))) → b(b(b(c(a(z0)))))
b(c(z0)) → a(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:none
Compound Symbols:none
(15) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(16) BOUNDS(O(1), O(1))