(0) Obligation:
The Runtime Complexity (full) of the given
CpxTRS could be proven to be
BOUNDS(1, n^1).
The TRS R consists of the following rules:
f(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
p(s(0)) → 0
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(X) → X
Rewrite Strategy: FULL
(1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)
Converted rc-obligation to irc-obligation.
As the TRS is a non-duplicating overlay system, we have rc = irc.
(2) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, n^1).
The TRS R consists of the following rules:
f(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
p(s(0)) → 0
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(X) → X
Rewrite Strategy: INNERMOST
(3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3]
transitions:
00() → 0
cons0(0, 0) → 0
n__f0(0) → 0
s0(0) → 0
f0(0) → 1
p0(0) → 2
activate0(0) → 3
01() → 4
01() → 7
s1(7) → 6
n__f1(6) → 5
cons1(4, 5) → 1
s1(7) → 9
p1(9) → 8
f1(8) → 1
01() → 2
n__f1(0) → 1
f1(0) → 3
cons1(4, 5) → 3
f1(8) → 3
02() → 8
n__f2(8) → 1
n__f2(0) → 3
n__f2(8) → 3
02() → 10
02() → 13
s2(13) → 12
n__f2(12) → 11
cons2(10, 11) → 1
cons2(10, 11) → 3
0 → 3
(4) BOUNDS(1, n^1)