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