(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(x1) → x1
a(b(x1)) → b(c(x1))
c(c(x1)) → b(a(c(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(z0)) → b(c(z0))
c(c(z0)) → b(a(c(a(a(z0)))))
Tuples:
A(b(z0)) → c2(C(z0))
C(c(z0)) → c3(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0))
S tuples:
A(b(z0)) → c2(C(z0))
C(c(z0)) → c3(A(c(a(a(z0)))), C(a(a(z0))), A(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(
c(
a(
a(
z0)))),
C(
a(
a(
z0))),
A(
a(
z0)),
A(
z0)) by
C(c(x0)) → c3(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0))
C(c(b(z0))) → c3(A(c(a(b(c(z0))))), C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(z0)) → b(c(z0))
c(c(z0)) → b(a(c(a(a(z0)))))
Tuples:
A(b(z0)) → c2(C(z0))
C(c(x0)) → c3(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0))
C(c(b(z0))) → c3(A(c(a(b(c(z0))))), C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
S tuples:
A(b(z0)) → c2(C(z0))
C(c(x0)) → c3(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0))
C(c(b(z0))) → c3(A(c(a(b(c(z0))))), C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
K tuples:none
Defined Rule Symbols:
a, c
Defined Pair Symbols:
A, C
Compound Symbols:
c2, c3
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
C(
c(
x0)) →
c3(
A(
c(
a(
x0))),
C(
a(
a(
x0))),
A(
a(
x0)),
A(
x0)) by
C(c(z0)) → c3(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0))
C(c(b(z0))) → c3(A(c(b(c(z0)))), C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(x0)) → c3(C(a(a(x0))))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(z0)) → b(c(z0))
c(c(z0)) → b(a(c(a(a(z0)))))
Tuples:
A(b(z0)) → c2(C(z0))
C(c(b(z0))) → c3(A(c(a(b(c(z0))))), C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(z0)) → c3(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0))
C(c(b(z0))) → c3(A(c(b(c(z0)))), C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(x0)) → c3(C(a(a(x0))))
S tuples:
A(b(z0)) → c2(C(z0))
C(c(b(z0))) → c3(A(c(a(b(c(z0))))), C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(z0)) → c3(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0))
C(c(b(z0))) → c3(A(c(b(c(z0)))), C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(x0)) → c3(C(a(a(x0))))
K tuples:none
Defined Rule Symbols:
a, c
Defined Pair Symbols:
A, C
Compound Symbols:
c2, c3, c3
(7) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(z0)) → b(c(z0))
c(c(z0)) → b(a(c(a(a(z0)))))
Tuples:
A(b(z0)) → c2(C(z0))
C(c(b(z0))) → c3(A(c(a(b(c(z0))))), C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(x0)) → c3(C(a(a(x0))))
C(c(z0)) → c3(C(a(a(z0))), A(a(z0)), A(z0))
C(c(b(z0))) → c3(C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
S tuples:
A(b(z0)) → c2(C(z0))
C(c(b(z0))) → c3(A(c(a(b(c(z0))))), C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(x0)) → c3(C(a(a(x0))))
C(c(z0)) → c3(C(a(a(z0))), A(a(z0)), A(z0))
C(c(b(z0))) → c3(C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
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(
b(
z0))) →
c3(
A(
c(
a(
b(
c(
z0))))),
C(
a(
a(
b(
z0)))),
A(
a(
b(
z0))),
A(
b(
z0))) by
C(c(b(x0))) → c3(A(c(b(c(x0)))), C(a(a(b(x0)))), A(a(b(x0))), A(b(x0)))
C(c(b(x0))) → c3(A(c(b(c(c(x0))))), C(a(a(b(x0)))), A(a(b(x0))), A(b(x0)))
C(c(b(c(z0)))) → c3(A(c(a(b(b(a(c(a(a(z0))))))))), C(a(a(b(c(z0))))), A(a(b(c(z0)))), A(b(c(z0))))
C(c(b(x0))) → c3(C(a(a(b(x0)))), A(a(b(x0))), A(b(x0)))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(z0)) → b(c(z0))
c(c(z0)) → b(a(c(a(a(z0)))))
Tuples:
A(b(z0)) → c2(C(z0))
C(c(x0)) → c3(C(a(a(x0))))
C(c(z0)) → c3(C(a(a(z0))), A(a(z0)), A(z0))
C(c(b(z0))) → c3(C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(b(x0))) → c3(A(c(b(c(x0)))), C(a(a(b(x0)))), A(a(b(x0))), A(b(x0)))
C(c(b(x0))) → c3(A(c(b(c(c(x0))))), C(a(a(b(x0)))), A(a(b(x0))), A(b(x0)))
C(c(b(c(z0)))) → c3(A(c(a(b(b(a(c(a(a(z0))))))))), C(a(a(b(c(z0))))), A(a(b(c(z0)))), A(b(c(z0))))
S tuples:
A(b(z0)) → c2(C(z0))
C(c(x0)) → c3(C(a(a(x0))))
C(c(z0)) → c3(C(a(a(z0))), A(a(z0)), A(z0))
C(c(b(z0))) → c3(C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(b(x0))) → c3(A(c(b(c(x0)))), C(a(a(b(x0)))), A(a(b(x0))), A(b(x0)))
C(c(b(x0))) → c3(A(c(b(c(c(x0))))), C(a(a(b(x0)))), A(a(b(x0))), A(b(x0)))
C(c(b(c(z0)))) → c3(A(c(a(b(b(a(c(a(a(z0))))))))), C(a(a(b(c(z0))))), A(a(b(c(z0)))), A(b(c(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 2 trailing tuple parts
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(z0)) → b(c(z0))
c(c(z0)) → b(a(c(a(a(z0)))))
Tuples:
A(b(z0)) → c2(C(z0))
C(c(x0)) → c3(C(a(a(x0))))
C(c(z0)) → c3(C(a(a(z0))), A(a(z0)), A(z0))
C(c(b(z0))) → c3(C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(b(c(z0)))) → c3(A(c(a(b(b(a(c(a(a(z0))))))))), C(a(a(b(c(z0))))), A(a(b(c(z0)))), A(b(c(z0))))
S tuples:
A(b(z0)) → c2(C(z0))
C(c(x0)) → c3(C(a(a(x0))))
C(c(z0)) → c3(C(a(a(z0))), A(a(z0)), A(z0))
C(c(b(z0))) → c3(C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(b(c(z0)))) → c3(A(c(a(b(b(a(c(a(a(z0))))))))), C(a(a(b(c(z0))))), A(a(b(c(z0)))), A(b(c(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(
z0)) →
c2(
C(
z0)) by
A(b(c(y0))) → c2(C(c(y0)))
A(b(c(b(y0)))) → c2(C(c(b(y0))))
A(b(c(b(c(y0))))) → c2(C(c(b(c(y0)))))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(z0)) → b(c(z0))
c(c(z0)) → b(a(c(a(a(z0)))))
Tuples:
C(c(x0)) → c3(C(a(a(x0))))
C(c(z0)) → c3(C(a(a(z0))), A(a(z0)), A(z0))
C(c(b(z0))) → c3(C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(b(c(z0)))) → c3(A(c(a(b(b(a(c(a(a(z0))))))))), C(a(a(b(c(z0))))), A(a(b(c(z0)))), A(b(c(z0))))
A(b(c(y0))) → c2(C(c(y0)))
A(b(c(b(y0)))) → c2(C(c(b(y0))))
A(b(c(b(c(y0))))) → c2(C(c(b(c(y0)))))
S tuples:
C(c(x0)) → c3(C(a(a(x0))))
C(c(z0)) → c3(C(a(a(z0))), A(a(z0)), A(z0))
C(c(b(z0))) → c3(C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(b(c(z0)))) → c3(A(c(a(b(b(a(c(a(a(z0))))))))), C(a(a(b(c(z0))))), A(a(b(c(z0)))), A(b(c(z0))))
A(b(c(y0))) → c2(C(c(y0)))
A(b(c(b(y0)))) → c2(C(c(b(y0))))
A(b(c(b(c(y0))))) → c2(C(c(b(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(x0)) → c3(C(a(a(x0))))
C(c(z0)) → c3(C(a(a(z0))), A(a(z0)), A(z0))
C(c(b(z0))) → c3(C(a(a(b(z0)))), A(a(b(z0))), A(b(z0)))
C(c(b(c(z0)))) → c3(A(c(a(b(b(a(c(a(a(z0))))))))), C(a(a(b(c(z0))))), A(a(b(c(z0)))), A(b(c(z0))))
A(b(c(y0))) → c2(C(c(y0)))
A(b(c(b(y0)))) → c2(C(c(b(y0))))
A(b(c(b(c(y0))))) → c2(C(c(b(c(y0)))))
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(z0)) → b(c(z0))
c(c(z0)) → b(a(c(a(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))