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