(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(c(d(x))) → c(x)
u(b(d(d(x)))) → b(x)
v(a(a(x))) → u(v(x))
v(a(c(x))) → u(b(d(x)))
v(c(x)) → b(x)
w(a(a(x))) → u(w(x))
w(a(c(x))) → u(b(d(x)))
w(c(x)) → b(x)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(c(d(z0))) → c(z0)
u(b(d(d(z0)))) → b(z0)
v(a(a(z0))) → u(v(z0))
v(a(c(z0))) → u(b(d(z0)))
v(c(z0)) → b(z0)
w(a(a(z0))) → u(w(z0))
w(a(c(z0))) → u(b(d(z0)))
w(c(z0)) → b(z0)
Tuples:
A(c(d(z0))) → c1
U(b(d(d(z0)))) → c2
V(a(a(z0))) → c3(U(v(z0)), V(z0))
V(a(c(z0))) → c4(U(b(d(z0))))
V(c(z0)) → c5
W(a(a(z0))) → c6(U(w(z0)), W(z0))
W(a(c(z0))) → c7(U(b(d(z0))))
W(c(z0)) → c8
S tuples:
A(c(d(z0))) → c1
U(b(d(d(z0)))) → c2
V(a(a(z0))) → c3(U(v(z0)), V(z0))
V(a(c(z0))) → c4(U(b(d(z0))))
V(c(z0)) → c5
W(a(a(z0))) → c6(U(w(z0)), W(z0))
W(a(c(z0))) → c7(U(b(d(z0))))
W(c(z0)) → c8
K tuples:none
Defined Rule Symbols:
a, u, v, w
Defined Pair Symbols:
A, U, V, W
Compound Symbols:
c1, c2, c3, c4, c5, c6, c7, c8
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing nodes:
U(b(d(d(z0)))) → c2
W(a(c(z0))) → c7(U(b(d(z0))))
W(c(z0)) → c8
V(a(c(z0))) → c4(U(b(d(z0))))
A(c(d(z0))) → c1
V(c(z0)) → c5
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(c(d(z0))) → c(z0)
u(b(d(d(z0)))) → b(z0)
v(a(a(z0))) → u(v(z0))
v(a(c(z0))) → u(b(d(z0)))
v(c(z0)) → b(z0)
w(a(a(z0))) → u(w(z0))
w(a(c(z0))) → u(b(d(z0)))
w(c(z0)) → b(z0)
Tuples:
V(a(a(z0))) → c3(U(v(z0)), V(z0))
W(a(a(z0))) → c6(U(w(z0)), W(z0))
S tuples:
V(a(a(z0))) → c3(U(v(z0)), V(z0))
W(a(a(z0))) → c6(U(w(z0)), W(z0))
K tuples:none
Defined Rule Symbols:
a, u, v, w
Defined Pair Symbols:
V, W
Compound Symbols:
c3, c6
(5) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
V(a(a(z0))) → c3(U(v(z0)), V(z0))
W(a(a(z0))) → c6(U(w(z0)), W(z0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(c(d(z0))) → c(z0)
u(b(d(d(z0)))) → b(z0)
v(a(a(z0))) → u(v(z0))
v(a(c(z0))) → u(b(d(z0)))
v(c(z0)) → b(z0)
w(a(a(z0))) → u(w(z0))
w(a(c(z0))) → u(b(d(z0)))
w(c(z0)) → b(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, u, v, w
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)