(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:
plus#2(0, x12) → x12
plus#2(S(x4), x2) → S(plus#2(x4, x2))
fold#3(Nil) → 0
fold#3(Cons(x4, x2)) → plus#2(x4, fold#3(x2))
main(x1) → fold#3(x1)
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:
00() → 0
S0(0) → 0
Nil0() → 0
Cons0(0, 0) → 0
plus#20(0, 0) → 1
fold#30(0) → 2
main0(0) → 3
plus#21(0, 0) → 4
S1(4) → 1
01() → 2
fold#31(0) → 5
plus#21(0, 5) → 2
fold#31(0) → 3
plus#21(0, 5) → 4
S1(4) → 2
S1(4) → 4
01() → 3
01() → 5
plus#21(0, 5) → 3
plus#21(0, 5) → 5
S1(4) → 3
S1(4) → 5
0 → 1
0 → 4
5 → 2
5 → 3
5 → 4
(2) BOUNDS(1, n^1)