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