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