(0) 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:
add0(x', Cons(x, xs)) → add0(Cons(Cons(Nil, Nil), x'), xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
add0(x, Nil) → x
goal(x, y) → add0(x, y)
Rewrite Strategy: INNERMOST
(1) 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 1.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3]
transitions:
Cons0(0, 0) → 0
Nil0() → 0
True0() → 0
False0() → 0
add00(0, 0) → 1
notEmpty0(0) → 2
goal0(0, 0) → 3
Nil1() → 6
Nil1() → 7
Cons1(6, 7) → 5
Cons1(5, 0) → 4
add01(4, 0) → 1
True1() → 2
False1() → 2
add01(0, 0) → 3
Cons1(5, 4) → 4
add01(4, 0) → 3
0 → 1
0 → 3
4 → 1
4 → 3
(2) BOUNDS(1, n^1)