(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
g(S(x), y) → g(x, S(y))
f(y, S(x)) → f(S(y), x)
g(0, x2) → x2
f(x1, 0) → g(x1, 0)
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 2.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2]
transitions:
S0(0) → 0
00() → 0
g0(0, 0) → 1
f0(0, 0) → 2
S1(0) → 3
g1(0, 3) → 1
S1(0) → 4
f1(4, 0) → 2
01() → 5
g1(0, 5) → 2
S1(3) → 3
S1(5) → 3
g1(0, 3) → 2
S1(4) → 4
g1(4, 5) → 2
S2(5) → 6
g2(0, 6) → 2
g2(4, 6) → 2
S1(6) → 3
S2(6) → 6
0 → 1
3 → 1
3 → 2
5 → 2
6 → 2
(2) BOUNDS(1, n^1)