(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(a(b(x1))) → b(a(x1))
b(a(a(x1))) → a(a(a(b(x1))))
a(c(x1)) → c(b(x1))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(b(z0))) → b(a(z0))
a(c(z0)) → c(b(z0))
b(a(a(z0))) → a(a(a(b(z0))))
Tuples:
A(a(b(z0))) → c1(B(a(z0)), A(z0))
A(c(z0)) → c2(B(z0))
B(a(a(z0))) → c3(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0))
S tuples:
A(a(b(z0))) → c1(B(a(z0)), A(z0))
A(c(z0)) → c2(B(z0))
B(a(a(z0))) → c3(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c2, c3
(3) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
B(
a(
a(
z0))) →
c3(
A(
a(
a(
b(
z0)))),
A(
a(
b(
z0))),
A(
b(
z0)),
B(
z0)) by
B(a(a(z0))) → c3(A(b(a(z0))), A(a(b(z0))), A(b(z0)), B(z0))
B(a(a(a(a(z0))))) → c3(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(b(z0))) → b(a(z0))
a(c(z0)) → c(b(z0))
b(a(a(z0))) → a(a(a(b(z0))))
Tuples:
A(a(b(z0))) → c1(B(a(z0)), A(z0))
A(c(z0)) → c2(B(z0))
B(a(a(z0))) → c3(A(b(a(z0))), A(a(b(z0))), A(b(z0)), B(z0))
B(a(a(a(a(z0))))) → c3(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))))
S tuples:
A(a(b(z0))) → c1(B(a(z0)), A(z0))
A(c(z0)) → c2(B(z0))
B(a(a(z0))) → c3(A(b(a(z0))), A(a(b(z0))), A(b(z0)), B(z0))
B(a(a(a(a(z0))))) → c3(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c2, c3
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
B(
a(
a(
z0))) →
c3(
A(
b(
a(
z0))),
A(
a(
b(
z0))),
A(
b(
z0)),
B(
z0)) by
B(a(a(a(z0)))) → c3(A(a(a(a(b(z0))))), A(a(b(a(z0)))), A(b(a(z0))), B(a(z0)))
B(a(a(x0))) → c3(A(a(b(x0))))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(b(z0))) → b(a(z0))
a(c(z0)) → c(b(z0))
b(a(a(z0))) → a(a(a(b(z0))))
Tuples:
A(a(b(z0))) → c1(B(a(z0)), A(z0))
A(c(z0)) → c2(B(z0))
B(a(a(a(a(z0))))) → c3(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))))
B(a(a(a(z0)))) → c3(A(a(a(a(b(z0))))), A(a(b(a(z0)))), A(b(a(z0))), B(a(z0)))
B(a(a(x0))) → c3(A(a(b(x0))))
S tuples:
A(a(b(z0))) → c1(B(a(z0)), A(z0))
A(c(z0)) → c2(B(z0))
B(a(a(a(a(z0))))) → c3(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))))
B(a(a(a(z0)))) → c3(A(a(a(a(b(z0))))), A(a(b(a(z0)))), A(b(a(z0))), B(a(z0)))
B(a(a(x0))) → c3(A(a(b(x0))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c2, c3, c3
(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
B(
a(
a(
a(
a(
z0))))) →
c3(
A(
a(
a(
a(
a(
a(
b(
z0))))))),
A(
a(
b(
a(
a(
z0))))),
A(
b(
a(
a(
z0)))),
B(
a(
a(
z0)))) by
B(a(a(a(a(z0))))) → c3(A(a(a(a(b(a(z0)))))), A(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))))
B(a(a(a(a(a(a(z0))))))) → c3(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(z0))))))), A(b(a(a(a(a(z0)))))), B(a(a(a(a(z0))))))
B(a(a(a(a(x0))))) → c3(A(a(b(a(a(x0))))))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(b(z0))) → b(a(z0))
a(c(z0)) → c(b(z0))
b(a(a(z0))) → a(a(a(b(z0))))
Tuples:
A(a(b(z0))) → c1(B(a(z0)), A(z0))
A(c(z0)) → c2(B(z0))
B(a(a(a(z0)))) → c3(A(a(a(a(b(z0))))), A(a(b(a(z0)))), A(b(a(z0))), B(a(z0)))
B(a(a(x0))) → c3(A(a(b(x0))))
B(a(a(a(a(z0))))) → c3(A(a(a(a(b(a(z0)))))), A(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))))
B(a(a(a(a(a(a(z0))))))) → c3(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(z0))))))), A(b(a(a(a(a(z0)))))), B(a(a(a(a(z0))))))
B(a(a(a(a(x0))))) → c3(A(a(b(a(a(x0))))))
S tuples:
A(a(b(z0))) → c1(B(a(z0)), A(z0))
A(c(z0)) → c2(B(z0))
B(a(a(a(z0)))) → c3(A(a(a(a(b(z0))))), A(a(b(a(z0)))), A(b(a(z0))), B(a(z0)))
B(a(a(x0))) → c3(A(a(b(x0))))
B(a(a(a(a(z0))))) → c3(A(a(a(a(b(a(z0)))))), A(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))))
B(a(a(a(a(a(a(z0))))))) → c3(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(z0))))))), A(b(a(a(a(a(z0)))))), B(a(a(a(a(z0))))))
B(a(a(a(a(x0))))) → c3(A(a(b(a(a(x0))))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c2, c3, c3
(9) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
A(
c(
z0)) →
c2(
B(
z0)) by
A(c(a(a(a(y0))))) → c2(B(a(a(a(y0)))))
A(c(a(a(y0)))) → c2(B(a(a(y0))))
A(c(a(a(a(a(y0)))))) → c2(B(a(a(a(a(y0))))))
A(c(a(a(a(a(a(a(y0)))))))) → c2(B(a(a(a(a(a(a(y0))))))))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(b(z0))) → b(a(z0))
a(c(z0)) → c(b(z0))
b(a(a(z0))) → a(a(a(b(z0))))
Tuples:
A(a(b(z0))) → c1(B(a(z0)), A(z0))
B(a(a(a(z0)))) → c3(A(a(a(a(b(z0))))), A(a(b(a(z0)))), A(b(a(z0))), B(a(z0)))
B(a(a(x0))) → c3(A(a(b(x0))))
B(a(a(a(a(z0))))) → c3(A(a(a(a(b(a(z0)))))), A(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))))
B(a(a(a(a(a(a(z0))))))) → c3(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(z0))))))), A(b(a(a(a(a(z0)))))), B(a(a(a(a(z0))))))
B(a(a(a(a(x0))))) → c3(A(a(b(a(a(x0))))))
A(c(a(a(a(y0))))) → c2(B(a(a(a(y0)))))
A(c(a(a(y0)))) → c2(B(a(a(y0))))
A(c(a(a(a(a(y0)))))) → c2(B(a(a(a(a(y0))))))
A(c(a(a(a(a(a(a(y0)))))))) → c2(B(a(a(a(a(a(a(y0))))))))
S tuples:
A(a(b(z0))) → c1(B(a(z0)), A(z0))
B(a(a(a(z0)))) → c3(A(a(a(a(b(z0))))), A(a(b(a(z0)))), A(b(a(z0))), B(a(z0)))
B(a(a(x0))) → c3(A(a(b(x0))))
B(a(a(a(a(z0))))) → c3(A(a(a(a(b(a(z0)))))), A(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))))
B(a(a(a(a(a(a(z0))))))) → c3(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(z0))))))), A(b(a(a(a(a(z0)))))), B(a(a(a(a(z0))))))
B(a(a(a(a(x0))))) → c3(A(a(b(a(a(x0))))))
A(c(a(a(a(y0))))) → c2(B(a(a(a(y0)))))
A(c(a(a(y0)))) → c2(B(a(a(y0))))
A(c(a(a(a(a(y0)))))) → c2(B(a(a(a(a(y0))))))
A(c(a(a(a(a(a(a(y0)))))))) → c2(B(a(a(a(a(a(a(y0))))))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c3, c3, c2
(11) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(a(b(z0))) → c1(B(a(z0)), A(z0))
B(a(a(a(z0)))) → c3(A(a(a(a(b(z0))))), A(a(b(a(z0)))), A(b(a(z0))), B(a(z0)))
B(a(a(x0))) → c3(A(a(b(x0))))
B(a(a(a(a(z0))))) → c3(A(a(a(a(b(a(z0)))))), A(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))))
B(a(a(a(a(a(a(z0))))))) → c3(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(z0))))))), A(b(a(a(a(a(z0)))))), B(a(a(a(a(z0))))))
B(a(a(a(a(x0))))) → c3(A(a(b(a(a(x0))))))
A(c(a(a(a(y0))))) → c2(B(a(a(a(y0)))))
A(c(a(a(y0)))) → c2(B(a(a(y0))))
A(c(a(a(a(a(y0)))))) → c2(B(a(a(a(a(y0))))))
A(c(a(a(a(a(a(a(y0)))))))) → c2(B(a(a(a(a(a(a(y0))))))))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(b(z0))) → b(a(z0))
a(c(z0)) → c(b(z0))
b(a(a(z0))) → a(a(a(b(z0))))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:none
Compound Symbols:none
(13) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(14) BOUNDS(O(1), O(1))