(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:
+(0, y) → y
+(s(x), 0) → s(x)
+(s(x), s(y)) → s(+(s(x), +(y, 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 3.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1]
transitions:
00() → 0
s0(0) → 0
+0(0, 0) → 1
s1(0) → 1
s1(0) → 3
01() → 5
+1(0, 5) → 4
+1(3, 4) → 2
s1(2) → 1
s1(0) → 4
s2(0) → 2
s2(0) → 7
02() → 9
+2(0, 9) → 8
+2(7, 8) → 6
s2(6) → 2
s1(0) → 8
s3(0) → 6
s2(6) → 6
0 → 1
5 → 4
9 → 8
(2) BOUNDS(1, n^1)