(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(a(x1)) → a(b(x1))
a(c(x1)) → b(x1)
b(b(x1)) → c(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(z0)) → a(b(z0))
a(c(z0)) → b(z0)
b(b(z0)) → c(b(a(z0)))
Tuples:
A(a(z0)) → c1(A(b(z0)), B(z0))
A(c(z0)) → c2(B(z0))
B(b(z0)) → c3(B(a(z0)), A(z0))
S tuples:
A(a(z0)) → c1(A(b(z0)), B(z0))
A(c(z0)) → c2(B(z0))
B(b(z0)) → c3(B(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c2, c3
(3) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
A(
a(
z0)) →
c1(
A(
b(
z0)),
B(
z0)) by
A(a(b(z0))) → c1(A(c(b(a(z0)))), B(b(z0)))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(z0)) → a(b(z0))
a(c(z0)) → b(z0)
b(b(z0)) → c(b(a(z0)))
Tuples:
A(c(z0)) → c2(B(z0))
B(b(z0)) → c3(B(a(z0)), A(z0))
A(a(b(z0))) → c1(A(c(b(a(z0)))), B(b(z0)))
S tuples:
A(c(z0)) → c2(B(z0))
B(b(z0)) → c3(B(a(z0)), A(z0))
A(a(b(z0))) → c1(A(c(b(a(z0)))), B(b(z0)))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c2, c3, c1
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
B(
b(
z0)) →
c3(
B(
a(
z0)),
A(
z0)) by
B(b(a(z0))) → c3(B(a(b(z0))), A(a(z0)))
B(b(c(z0))) → c3(B(b(z0)), A(c(z0)))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(z0)) → a(b(z0))
a(c(z0)) → b(z0)
b(b(z0)) → c(b(a(z0)))
Tuples:
A(c(z0)) → c2(B(z0))
A(a(b(z0))) → c1(A(c(b(a(z0)))), B(b(z0)))
B(b(a(z0))) → c3(B(a(b(z0))), A(a(z0)))
B(b(c(z0))) → c3(B(b(z0)), A(c(z0)))
S tuples:
A(c(z0)) → c2(B(z0))
A(a(b(z0))) → c1(A(c(b(a(z0)))), B(b(z0)))
B(b(a(z0))) → c3(B(a(b(z0))), A(a(z0)))
B(b(c(z0))) → c3(B(b(z0)), A(c(z0)))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c2, c1, c3
(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
B(
b(
a(
z0))) →
c3(
B(
a(
b(
z0))),
A(
a(
z0))) by
B(b(a(b(z0)))) → c3(B(a(c(b(a(z0))))), A(a(b(z0))))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(z0)) → a(b(z0))
a(c(z0)) → b(z0)
b(b(z0)) → c(b(a(z0)))
Tuples:
A(c(z0)) → c2(B(z0))
A(a(b(z0))) → c1(A(c(b(a(z0)))), B(b(z0)))
B(b(c(z0))) → c3(B(b(z0)), A(c(z0)))
B(b(a(b(z0)))) → c3(B(a(c(b(a(z0))))), A(a(b(z0))))
S tuples:
A(c(z0)) → c2(B(z0))
A(a(b(z0))) → c1(A(c(b(a(z0)))), B(b(z0)))
B(b(c(z0))) → c3(B(b(z0)), A(c(z0)))
B(b(a(b(z0)))) → c3(B(a(c(b(a(z0))))), A(a(b(z0))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c2, c1, c3
(9) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
A(
c(
z0)) →
c2(
B(
z0)) by
A(c(b(c(y0)))) → c2(B(b(c(y0))))
A(c(b(a(b(y0))))) → c2(B(b(a(b(y0)))))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(z0)) → a(b(z0))
a(c(z0)) → b(z0)
b(b(z0)) → c(b(a(z0)))
Tuples:
A(a(b(z0))) → c1(A(c(b(a(z0)))), B(b(z0)))
B(b(c(z0))) → c3(B(b(z0)), A(c(z0)))
B(b(a(b(z0)))) → c3(B(a(c(b(a(z0))))), A(a(b(z0))))
A(c(b(c(y0)))) → c2(B(b(c(y0))))
A(c(b(a(b(y0))))) → c2(B(b(a(b(y0)))))
S tuples:
A(a(b(z0))) → c1(A(c(b(a(z0)))), B(b(z0)))
B(b(c(z0))) → c3(B(b(z0)), A(c(z0)))
B(b(a(b(z0)))) → c3(B(a(c(b(a(z0))))), A(a(b(z0))))
A(c(b(c(y0)))) → c2(B(b(c(y0))))
A(c(b(a(b(y0))))) → c2(B(b(a(b(y0)))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c3, c2
(11) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(a(b(z0))) → c1(A(c(b(a(z0)))), B(b(z0)))
B(b(c(z0))) → c3(B(b(z0)), A(c(z0)))
B(b(a(b(z0)))) → c3(B(a(c(b(a(z0))))), A(a(b(z0))))
A(c(b(c(y0)))) → c2(B(b(c(y0))))
A(c(b(a(b(y0))))) → c2(B(b(a(b(y0)))))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(z0)) → a(b(z0))
a(c(z0)) → b(z0)
b(b(z0)) → c(b(a(z0)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:none
Compound Symbols:none
(13) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(14) BOUNDS(O(1), O(1))