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