(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(x1) → b(x1)
a(a(x1)) → a(b(a(x1)))
a(b(x1)) → b(b(b(x1)))
a(a(a(x1))) → a(a(b(a(a(x1)))))
a(a(b(x1))) → a(b(b(a(b(x1)))))
a(b(a(x1))) → b(a(b(b(a(x1)))))
a(b(b(x1))) → b(b(b(b(b(x1)))))
a(a(a(a(x1)))) → a(a(a(b(a(a(a(x1)))))))
a(a(a(b(x1)))) → a(a(b(b(a(a(b(x1)))))))
a(a(b(a(x1)))) → a(b(a(b(a(b(a(x1)))))))
a(a(b(b(x1)))) → a(b(b(b(a(b(b(x1)))))))
a(b(a(a(x1)))) → b(a(a(b(b(a(a(x1)))))))
a(b(a(b(x1)))) → b(a(b(b(b(a(b(x1)))))))
a(b(b(a(x1)))) → b(b(a(b(b(b(a(x1)))))))
a(b(b(b(x1)))) → b(b(b(b(b(b(b(x1)))))))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → b(z0)
a(a(z0)) → a(b(a(z0)))
a(b(z0)) → b(b(b(z0)))
a(a(a(z0))) → a(a(b(a(a(z0)))))
a(a(b(z0))) → a(b(b(a(b(z0)))))
a(b(a(z0))) → b(a(b(b(a(z0)))))
a(b(b(z0))) → b(b(b(b(b(z0)))))
a(a(a(a(z0)))) → a(a(a(b(a(a(a(z0)))))))
a(a(a(b(z0)))) → a(a(b(b(a(a(b(z0)))))))
a(a(b(a(z0)))) → a(b(a(b(a(b(a(z0)))))))
a(a(b(b(z0)))) → a(b(b(b(a(b(b(z0)))))))
a(b(a(a(z0)))) → b(a(a(b(b(a(a(z0)))))))
a(b(a(b(z0)))) → b(a(b(b(b(a(b(z0)))))))
a(b(b(a(z0)))) → b(b(a(b(b(b(a(z0)))))))
a(b(b(b(z0)))) → b(b(b(b(b(b(b(z0)))))))
Tuples:
A(a(z0)) → c1(A(b(a(z0))), A(z0))
A(a(a(z0))) → c3(A(a(b(a(a(z0))))), A(b(a(a(z0)))), A(a(z0)), A(z0))
A(a(b(z0))) → c4(A(b(b(a(b(z0))))), A(b(z0)))
A(b(a(z0))) → c5(A(b(b(a(z0)))), A(z0))
A(a(a(a(z0)))) → c7(A(a(a(b(a(a(a(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), A(a(a(z0))), A(a(z0)), A(z0))
A(a(a(b(z0)))) → c8(A(a(b(b(a(a(b(z0))))))), A(b(b(a(a(b(z0)))))), A(a(b(z0))), A(b(z0)))
A(a(b(a(z0)))) → c9(A(b(a(b(a(b(a(z0))))))), A(b(a(b(a(z0))))), A(b(a(z0))), A(z0))
A(a(b(b(z0)))) → c10(A(b(b(b(a(b(b(z0))))))), A(b(b(z0))))
A(b(a(a(z0)))) → c11(A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), A(a(z0)), A(z0))
A(b(a(b(z0)))) → c12(A(b(b(b(a(b(z0)))))), A(b(z0)))
A(b(b(a(z0)))) → c13(A(b(b(b(a(z0))))), A(z0))
S tuples:
A(a(z0)) → c1(A(b(a(z0))), A(z0))
A(a(a(z0))) → c3(A(a(b(a(a(z0))))), A(b(a(a(z0)))), A(a(z0)), A(z0))
A(a(b(z0))) → c4(A(b(b(a(b(z0))))), A(b(z0)))
A(b(a(z0))) → c5(A(b(b(a(z0)))), A(z0))
A(a(a(a(z0)))) → c7(A(a(a(b(a(a(a(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), A(a(a(z0))), A(a(z0)), A(z0))
A(a(a(b(z0)))) → c8(A(a(b(b(a(a(b(z0))))))), A(b(b(a(a(b(z0)))))), A(a(b(z0))), A(b(z0)))
A(a(b(a(z0)))) → c9(A(b(a(b(a(b(a(z0))))))), A(b(a(b(a(z0))))), A(b(a(z0))), A(z0))
A(a(b(b(z0)))) → c10(A(b(b(b(a(b(b(z0))))))), A(b(b(z0))))
A(b(a(a(z0)))) → c11(A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), A(a(z0)), A(z0))
A(b(a(b(z0)))) → c12(A(b(b(b(a(b(z0)))))), A(b(z0)))
A(b(b(a(z0)))) → c13(A(b(b(b(a(z0))))), A(z0))
K tuples:none
Defined Rule Symbols:
a
Defined Pair Symbols:
A
Compound Symbols:
c1, c3, c4, c5, c7, c8, c9, c10, c11, c12, c13
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(a(z0)) → c1(A(b(a(z0))), A(z0))
A(a(a(z0))) → c3(A(a(b(a(a(z0))))), A(b(a(a(z0)))), A(a(z0)), A(z0))
A(a(b(z0))) → c4(A(b(b(a(b(z0))))), A(b(z0)))
A(b(a(z0))) → c5(A(b(b(a(z0)))), A(z0))
A(a(a(a(z0)))) → c7(A(a(a(b(a(a(a(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), A(a(a(z0))), A(a(z0)), A(z0))
A(a(a(b(z0)))) → c8(A(a(b(b(a(a(b(z0))))))), A(b(b(a(a(b(z0)))))), A(a(b(z0))), A(b(z0)))
A(a(b(a(z0)))) → c9(A(b(a(b(a(b(a(z0))))))), A(b(a(b(a(z0))))), A(b(a(z0))), A(z0))
A(a(b(b(z0)))) → c10(A(b(b(b(a(b(b(z0))))))), A(b(b(z0))))
A(b(a(a(z0)))) → c11(A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), A(a(z0)), A(z0))
A(b(a(b(z0)))) → c12(A(b(b(b(a(b(z0)))))), A(b(z0)))
A(b(b(a(z0)))) → c13(A(b(b(b(a(z0))))), A(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → b(z0)
a(a(z0)) → a(b(a(z0)))
a(b(z0)) → b(b(b(z0)))
a(a(a(z0))) → a(a(b(a(a(z0)))))
a(a(b(z0))) → a(b(b(a(b(z0)))))
a(b(a(z0))) → b(a(b(b(a(z0)))))
a(b(b(z0))) → b(b(b(b(b(z0)))))
a(a(a(a(z0)))) → a(a(a(b(a(a(a(z0)))))))
a(a(a(b(z0)))) → a(a(b(b(a(a(b(z0)))))))
a(a(b(a(z0)))) → a(b(a(b(a(b(a(z0)))))))
a(a(b(b(z0)))) → a(b(b(b(a(b(b(z0)))))))
a(b(a(a(z0)))) → b(a(a(b(b(a(a(z0)))))))
a(b(a(b(z0)))) → b(a(b(b(b(a(b(z0)))))))
a(b(b(a(z0)))) → b(b(a(b(b(b(a(z0)))))))
a(b(b(b(z0)))) → b(b(b(b(b(b(b(z0)))))))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))