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