(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(c(x1)) → c(b(x1))
a(x1) → b(b(b(x1)))
b(c(b(x1))) → 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(c(z0)) → c(b(z0))
a(z0) → b(b(b(z0)))
b(c(b(z0))) → a(c(z0))
Tuples:
A(c(z0)) → c1(B(z0))
A(z0) → c2(B(b(b(z0))), B(b(z0)), B(z0))
B(c(b(z0))) → c3(A(c(z0)))
S tuples:
A(c(z0)) → c1(B(z0))
A(z0) → c2(B(b(b(z0))), B(b(z0)), B(z0))
B(c(b(z0))) → c3(A(c(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(
z0) →
c2(
B(
b(
b(
z0))),
B(
b(
z0)),
B(
z0)) by
A(c(b(z0))) → c2(B(b(a(c(z0)))), B(b(c(b(z0)))), B(c(b(z0))))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(c(z0)) → c(b(z0))
a(z0) → b(b(b(z0)))
b(c(b(z0))) → a(c(z0))
Tuples:
A(c(z0)) → c1(B(z0))
B(c(b(z0))) → c3(A(c(z0)))
A(c(b(z0))) → c2(B(b(a(c(z0)))), B(b(c(b(z0)))), B(c(b(z0))))
S tuples:
A(c(z0)) → c1(B(z0))
B(c(b(z0))) → c3(A(c(z0)))
A(c(b(z0))) → c2(B(b(a(c(z0)))), B(b(c(b(z0)))), B(c(b(z0))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c3, c2
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
A(
c(
b(
z0))) →
c2(
B(
b(
a(
c(
z0)))),
B(
b(
c(
b(
z0)))),
B(
c(
b(
z0)))) by
A(c(b(z0))) → c2(B(b(c(b(z0)))), B(b(c(b(z0)))), B(c(b(z0))))
A(c(b(x0))) → c2(B(b(b(b(b(c(x0)))))), B(b(c(b(x0)))), B(c(b(x0))))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(c(z0)) → c(b(z0))
a(z0) → b(b(b(z0)))
b(c(b(z0))) → a(c(z0))
Tuples:
A(c(z0)) → c1(B(z0))
B(c(b(z0))) → c3(A(c(z0)))
A(c(b(z0))) → c2(B(b(c(b(z0)))), B(b(c(b(z0)))), B(c(b(z0))))
A(c(b(x0))) → c2(B(b(b(b(b(c(x0)))))), B(b(c(b(x0)))), B(c(b(x0))))
S tuples:
A(c(z0)) → c1(B(z0))
B(c(b(z0))) → c3(A(c(z0)))
A(c(b(z0))) → c2(B(b(c(b(z0)))), B(b(c(b(z0)))), B(c(b(z0))))
A(c(b(x0))) → c2(B(b(b(b(b(c(x0)))))), B(b(c(b(x0)))), B(c(b(x0))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c3, c2
(7) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
A(
c(
z0)) →
c1(
B(
z0)) by
A(c(c(b(y0)))) → c1(B(c(b(y0))))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(c(z0)) → c(b(z0))
a(z0) → b(b(b(z0)))
b(c(b(z0))) → a(c(z0))
Tuples:
B(c(b(z0))) → c3(A(c(z0)))
A(c(b(z0))) → c2(B(b(c(b(z0)))), B(b(c(b(z0)))), B(c(b(z0))))
A(c(b(x0))) → c2(B(b(b(b(b(c(x0)))))), B(b(c(b(x0)))), B(c(b(x0))))
A(c(c(b(y0)))) → c1(B(c(b(y0))))
S tuples:
B(c(b(z0))) → c3(A(c(z0)))
A(c(b(z0))) → c2(B(b(c(b(z0)))), B(b(c(b(z0)))), B(c(b(z0))))
A(c(b(x0))) → c2(B(b(b(b(b(c(x0)))))), B(b(c(b(x0)))), B(c(b(x0))))
A(c(c(b(y0)))) → c1(B(c(b(y0))))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
B, A
Compound Symbols:
c3, c2, c1
(9) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
B(c(b(z0))) → c3(A(c(z0)))
A(c(b(z0))) → c2(B(b(c(b(z0)))), B(b(c(b(z0)))), B(c(b(z0))))
A(c(b(x0))) → c2(B(b(b(b(b(c(x0)))))), B(b(c(b(x0)))), B(c(b(x0))))
A(c(c(b(y0)))) → c1(B(c(b(y0))))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(c(z0)) → c(b(z0))
a(z0) → b(b(b(z0)))
b(c(b(z0))) → a(c(z0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:none
Compound Symbols:none
(11) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(12) BOUNDS(O(1), O(1))