(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(a(b(b(x1)))) → b(b(b(a(a(a(x1))))))
a(c(x1)) → c(a(x1))
c(b(x1)) → b(c(x1))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(b(b(z0)))) → b(b(b(a(a(a(z0))))))
a(c(z0)) → c(a(z0))
c(b(z0)) → b(c(z0))
Tuples:
A(a(b(b(z0)))) → c1(A(a(a(z0))), A(a(z0)), A(z0))
A(c(z0)) → c2(C(a(z0)), A(z0))
C(b(z0)) → c3(C(z0))
S tuples:
A(a(b(b(z0)))) → c1(A(a(a(z0))), A(a(z0)), A(z0))
A(c(z0)) → c2(C(a(z0)), A(z0))
C(b(z0)) → c3(C(z0))
K tuples:none
Defined Rule Symbols:
a, c
Defined Pair Symbols:
A, C
Compound Symbols:
c1, c2, c3
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(a(b(b(z0)))) → c1(A(a(a(z0))), A(a(z0)), A(z0))
A(c(z0)) → c2(C(a(z0)), A(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(b(b(z0)))) → b(b(b(a(a(a(z0))))))
a(c(z0)) → c(a(z0))
c(b(z0)) → b(c(z0))
Tuples:
C(b(z0)) → c3(C(z0))
S tuples:
C(b(z0)) → c3(C(z0))
K tuples:none
Defined Rule Symbols:
a, c
Defined Pair Symbols:
C
Compound Symbols:
c3
(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
C(b(z0)) → c3(C(z0))
We considered the (Usable) Rules:none
And the Tuples:
C(b(z0)) → c3(C(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(C(x1)) = [5]x1
POL(b(x1)) = [1] + x1
POL(c3(x1)) = x1
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(b(b(z0)))) → b(b(b(a(a(a(z0))))))
a(c(z0)) → c(a(z0))
c(b(z0)) → b(c(z0))
Tuples:
C(b(z0)) → c3(C(z0))
S tuples:none
K tuples:
C(b(z0)) → c3(C(z0))
Defined Rule Symbols:
a, c
Defined Pair Symbols:
C
Compound Symbols:
c3
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))