(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
u(a(x)) → x
v(b(x)) → x
w(c(x)) → x
a(u(x)) → x
b(v(x)) → x
c(w(x)) → x
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(b(z0)) → b(a(a(z0)))
a(u(z0)) → z0
b(c(z0)) → c(b(b(z0)))
b(v(z0)) → z0
c(a(z0)) → a(c(c(z0)))
c(w(z0)) → z0
u(a(z0)) → z0
v(b(z0)) → z0
w(c(z0)) → z0
Tuples:
A(b(z0)) → c1(B(a(a(z0))), A(a(z0)), A(z0))
B(c(z0)) → c3(C(b(b(z0))), B(b(z0)), B(z0))
C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0))
S tuples:
A(b(z0)) → c1(B(a(a(z0))), A(a(z0)), A(z0))
B(c(z0)) → c3(C(b(b(z0))), B(b(z0)), B(z0))
C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0))
K tuples:none
Defined Rule Symbols:
a, b, c, u, v, w
Defined Pair Symbols:
A, B, C
Compound Symbols:
c1, c3, c5
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(b(z0)) → c1(B(a(a(z0))), A(a(z0)), A(z0))
B(c(z0)) → c3(C(b(b(z0))), B(b(z0)), B(z0))
C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(b(z0)) → b(a(a(z0)))
a(u(z0)) → z0
b(c(z0)) → c(b(b(z0)))
b(v(z0)) → z0
c(a(z0)) → a(c(c(z0)))
c(w(z0)) → z0
u(a(z0)) → z0
v(b(z0)) → z0
w(c(z0)) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, b, c, u, v, w
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))