(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(a(a(x1))) → a(b(c(c(c(x1)))))
c(b(x1)) → b(a(x1))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(a(z0))) → a(b(c(c(c(z0)))))
c(b(z0)) → b(a(z0))
Tuples:
A(a(a(z0))) → c1(A(b(c(c(c(z0))))), C(c(c(z0))), C(c(z0)), C(z0))
C(b(z0)) → c2(A(z0))
S tuples:
A(a(a(z0))) → c1(A(b(c(c(c(z0))))), C(c(c(z0))), C(c(z0)), C(z0))
C(b(z0)) → c2(A(z0))
K tuples:none
Defined Rule Symbols:
a, c
Defined Pair Symbols:
A, C
Compound Symbols:
c1, c2
(3) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(a(z0))) → a(b(c(c(c(z0)))))
c(b(z0)) → b(a(z0))
Tuples:
C(b(z0)) → c2(A(z0))
A(a(a(z0))) → c1(C(c(c(z0))), C(c(z0)), C(z0))
S tuples:
C(b(z0)) → c2(A(z0))
A(a(a(z0))) → c1(C(c(c(z0))), C(c(z0)), C(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
A(
a(
a(
z0))) →
c1(
C(
c(
c(
z0))),
C(
c(
z0)),
C(
z0)) by
A(a(a(b(z0)))) → c1(C(c(b(a(z0)))), C(c(b(z0))), C(b(z0)))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(a(z0))) → a(b(c(c(c(z0)))))
c(b(z0)) → b(a(z0))
Tuples:
C(b(z0)) → c2(A(z0))
A(a(a(b(z0)))) → c1(C(c(b(a(z0)))), C(c(b(z0))), C(b(z0)))
S tuples:
C(b(z0)) → c2(A(z0))
A(a(a(b(z0)))) → c1(C(c(b(a(z0)))), C(c(b(z0))), C(b(z0)))
K tuples:none
Defined Rule Symbols:
a, c
Defined Pair Symbols:
C, A
Compound Symbols:
c2, c1
(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
A(
a(
a(
b(
z0)))) →
c1(
C(
c(
b(
a(
z0)))),
C(
c(
b(
z0))),
C(
b(
z0))) by
A(a(a(b(x0)))) → c1(C(b(a(a(x0)))), C(c(b(x0))), C(b(x0)))
A(a(a(b(a(a(z0)))))) → c1(C(c(b(a(b(c(c(c(z0)))))))), C(c(b(a(a(z0))))), C(b(a(a(z0)))))
A(a(a(b(x0)))) → c1(C(c(b(x0))), C(b(x0)))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(a(z0))) → a(b(c(c(c(z0)))))
c(b(z0)) → b(a(z0))
Tuples:
C(b(z0)) → c2(A(z0))
A(a(a(b(x0)))) → c1(C(b(a(a(x0)))), C(c(b(x0))), C(b(x0)))
A(a(a(b(a(a(z0)))))) → c1(C(c(b(a(b(c(c(c(z0)))))))), C(c(b(a(a(z0))))), C(b(a(a(z0)))))
A(a(a(b(x0)))) → c1(C(c(b(x0))), C(b(x0)))
S tuples:
C(b(z0)) → c2(A(z0))
A(a(a(b(x0)))) → c1(C(b(a(a(x0)))), C(c(b(x0))), C(b(x0)))
A(a(a(b(a(a(z0)))))) → c1(C(c(b(a(b(c(c(c(z0)))))))), C(c(b(a(a(z0))))), C(b(a(a(z0)))))
A(a(a(b(x0)))) → c1(C(c(b(x0))), C(b(x0)))
K tuples:none
Defined Rule Symbols:
a, c
Defined Pair Symbols:
C, A
Compound Symbols:
c2, c1, c1
(9) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
C(
b(
z0)) →
c2(
A(
z0)) by
C(b(a(a(b(y0))))) → c2(A(a(a(b(y0)))))
C(b(a(a(b(a(a(y0))))))) → c2(A(a(a(b(a(a(y0)))))))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(a(z0))) → a(b(c(c(c(z0)))))
c(b(z0)) → b(a(z0))
Tuples:
A(a(a(b(x0)))) → c1(C(b(a(a(x0)))), C(c(b(x0))), C(b(x0)))
A(a(a(b(a(a(z0)))))) → c1(C(c(b(a(b(c(c(c(z0)))))))), C(c(b(a(a(z0))))), C(b(a(a(z0)))))
A(a(a(b(x0)))) → c1(C(c(b(x0))), C(b(x0)))
C(b(a(a(b(y0))))) → c2(A(a(a(b(y0)))))
C(b(a(a(b(a(a(y0))))))) → c2(A(a(a(b(a(a(y0)))))))
S tuples:
A(a(a(b(x0)))) → c1(C(b(a(a(x0)))), C(c(b(x0))), C(b(x0)))
A(a(a(b(a(a(z0)))))) → c1(C(c(b(a(b(c(c(c(z0)))))))), C(c(b(a(a(z0))))), C(b(a(a(z0)))))
A(a(a(b(x0)))) → c1(C(c(b(x0))), C(b(x0)))
C(b(a(a(b(y0))))) → c2(A(a(a(b(y0)))))
C(b(a(a(b(a(a(y0))))))) → c2(A(a(a(b(a(a(y0)))))))
K tuples:none
Defined Rule Symbols:
a, c
Defined Pair Symbols:
A, C
Compound Symbols:
c1, c1, c2
(11) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(a(a(b(x0)))) → c1(C(b(a(a(x0)))), C(c(b(x0))), C(b(x0)))
A(a(a(b(a(a(z0)))))) → c1(C(c(b(a(b(c(c(c(z0)))))))), C(c(b(a(a(z0))))), C(b(a(a(z0)))))
A(a(a(b(x0)))) → c1(C(c(b(x0))), C(b(x0)))
C(b(a(a(b(y0))))) → c2(A(a(a(b(y0)))))
C(b(a(a(b(a(a(y0))))))) → c2(A(a(a(b(a(a(y0)))))))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(a(z0))) → a(b(c(c(c(z0)))))
c(b(z0)) → b(a(z0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, c
Defined Pair Symbols:none
Compound Symbols:none
(13) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(14) BOUNDS(O(1), O(1))