(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(x1) → x1
a(b(b(x1))) → b(b(b(c(x1))))
b(c(x1)) → a(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(z0))))
b(c(z0)) → a(a(z0))
Tuples:
A(b(b(z0))) → c2(B(b(b(c(z0)))), B(b(c(z0))), B(c(z0)))
B(c(z0)) → c3(A(a(z0)), A(z0))
S tuples:
A(b(b(z0))) → c2(B(b(b(c(z0)))), B(b(c(z0))), B(c(z0)))
B(c(z0)) → c3(A(a(z0)), 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(
z0)))),
B(
b(
c(
z0))),
B(
c(
z0))) by
A(b(b(z0))) → c2(B(b(a(a(z0)))), B(b(c(z0))), B(c(z0)))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(b(z0))) → b(b(b(c(z0))))
b(c(z0)) → a(a(z0))
Tuples:
B(c(z0)) → c3(A(a(z0)), A(z0))
A(b(b(z0))) → c2(B(b(a(a(z0)))), B(b(c(z0))), B(c(z0)))
S tuples:
B(c(z0)) → c3(A(a(z0)), A(z0))
A(b(b(z0))) → c2(B(b(a(a(z0)))), B(b(c(z0))), B(c(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
B(
c(
z0)) →
c3(
A(
a(
z0)),
A(
z0)) by
B(c(z0)) → c3(A(z0), A(z0))
B(c(b(b(z0)))) → c3(A(b(b(b(c(z0))))), A(b(b(z0))))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(b(z0))) → b(b(b(c(z0))))
b(c(z0)) → a(a(z0))
Tuples:
A(b(b(z0))) → c2(B(b(a(a(z0)))), B(b(c(z0))), B(c(z0)))
B(c(z0)) → c3(A(z0), A(z0))
B(c(b(b(z0)))) → c3(A(b(b(b(c(z0))))), A(b(b(z0))))
S tuples:
A(b(b(z0))) → c2(B(b(a(a(z0)))), B(b(c(z0))), B(c(z0)))
B(c(z0)) → c3(A(z0), A(z0))
B(c(b(b(z0)))) → c3(A(b(b(b(c(z0))))), A(b(b(z0))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c2, c3
(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
A(
b(
b(
z0))) →
c2(
B(
b(
a(
a(
z0)))),
B(
b(
c(
z0))),
B(
c(
z0))) by
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(x0))), B(c(x0)))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(z0))))))), B(b(c(b(b(z0))))), B(c(b(b(z0)))))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(b(z0))) → b(b(b(c(z0))))
b(c(z0)) → a(a(z0))
Tuples:
B(c(z0)) → c3(A(z0), A(z0))
B(c(b(b(z0)))) → c3(A(b(b(b(c(z0))))), A(b(b(z0))))
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(x0))), B(c(x0)))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(z0))))))), B(b(c(b(b(z0))))), B(c(b(b(z0)))))
S tuples:
B(c(z0)) → c3(A(z0), A(z0))
B(c(b(b(z0)))) → c3(A(b(b(b(c(z0))))), A(b(b(z0))))
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(x0))), B(c(x0)))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(z0))))))), B(b(c(b(b(z0))))), B(c(b(b(z0)))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
B, A
Compound Symbols:
c3, c2
(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
B(
c(
b(
b(
z0)))) →
c3(
A(
b(
b(
b(
c(
z0))))),
A(
b(
b(
z0)))) by
B(c(b(b(z0)))) → c3(A(b(b(a(a(z0))))), A(b(b(z0))))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(b(z0))) → b(b(b(c(z0))))
b(c(z0)) → a(a(z0))
Tuples:
B(c(z0)) → c3(A(z0), A(z0))
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(x0))), B(c(x0)))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(z0))))))), B(b(c(b(b(z0))))), B(c(b(b(z0)))))
B(c(b(b(z0)))) → c3(A(b(b(a(a(z0))))), A(b(b(z0))))
S tuples:
B(c(z0)) → c3(A(z0), A(z0))
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(x0))), B(c(x0)))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(z0))))))), B(b(c(b(b(z0))))), B(c(b(b(z0)))))
B(c(b(b(z0)))) → c3(A(b(b(a(a(z0))))), A(b(b(z0))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
B, A
Compound Symbols:
c3, c2
(11) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
B(
c(
z0)) →
c3(
A(
z0),
A(
z0)) by
B(c(b(b(y0)))) → c3(A(b(b(y0))), A(b(b(y0))))
B(c(b(b(b(b(y0)))))) → c3(A(b(b(b(b(y0))))), A(b(b(b(b(y0))))))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(b(z0))) → b(b(b(c(z0))))
b(c(z0)) → a(a(z0))
Tuples:
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(x0))), B(c(x0)))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(z0))))))), B(b(c(b(b(z0))))), B(c(b(b(z0)))))
B(c(b(b(z0)))) → c3(A(b(b(a(a(z0))))), A(b(b(z0))))
B(c(b(b(y0)))) → c3(A(b(b(y0))), A(b(b(y0))))
B(c(b(b(b(b(y0)))))) → c3(A(b(b(b(b(y0))))), A(b(b(b(b(y0))))))
S tuples:
A(b(b(x0))) → c2(B(b(a(x0))), B(b(c(x0))), B(c(x0)))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(z0))))))), B(b(c(b(b(z0))))), B(c(b(b(z0)))))
B(c(b(b(z0)))) → c3(A(b(b(a(a(z0))))), A(b(b(z0))))
B(c(b(b(y0)))) → c3(A(b(b(y0))), A(b(b(y0))))
B(c(b(b(b(b(y0)))))) → c3(A(b(b(b(b(y0))))), A(b(b(b(b(y0))))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
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(x0))), B(c(x0)))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(b(c(z0))))))), B(b(c(b(b(z0))))), B(c(b(b(z0)))))
B(c(b(b(z0)))) → c3(A(b(b(a(a(z0))))), A(b(b(z0))))
B(c(b(b(y0)))) → c3(A(b(b(y0))), A(b(b(y0))))
B(c(b(b(b(b(y0)))))) → c3(A(b(b(b(b(y0))))), A(b(b(b(b(y0))))))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(b(z0))) → b(b(b(c(z0))))
b(c(z0)) → a(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))