(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:
merge(x, nil) → x
merge(nil, y) → y
merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v))
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]
transitions:
nil0() → 0
++0(0, 0) → 0
u0() → 0
v0() → 0
merge0(0, 0) → 1
u1() → 4
v1() → 5
++1(4, 5) → 3
merge1(0, 3) → 2
++1(0, 2) → 1
++1(0, 0) → 7
v1() → 8
merge1(7, 8) → 6
++1(4, 6) → 1
++1(0, 2) → 2
++1(4, 6) → 2
0 → 1
3 → 2
(2) BOUNDS(1, n^1)