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