(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) CpxTrsMatchBoundsProof (EQUIVALENT transformation)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1.
The certificate found is represented by the following graph.
Start state: 1
Accept states: [2]
Transitions:
1→2[v_1|0, a_1|0, u_1|0, w_1|0, b_1|1, c_1|1]
2→2[c_1|0, b_1|0, d_1|0]
(4) BOUNDS(1, n^1)
(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(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:
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
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
U(b(d(d(z0)))) → c3
V(c(z0)) → c1
A(c(d(z0))) → c2
W(c(z0)) → c4
(8) 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
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(1, 1)