### (0) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

The TRS R consists of the following rules:

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
activate(X) → X

Rewrite Strategy: FULL

### (1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.

### (2) 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:

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
activate(X) → X

Rewrite Strategy: INNERMOST

### (3) 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, 2, 3, 4]
transitions:
tt0() → 0
s0(0) → 0
00() → 0
U110(0, 0, 0) → 1
U120(0, 0, 0) → 2
plus0(0, 0) → 3
activate0(0) → 4
tt1() → 5
activate1(0) → 6
activate1(0) → 7
U121(5, 6, 7) → 1
activate1(0) → 9
activate1(0) → 10
plus1(9, 10) → 8
s1(8) → 2
tt1() → 11
U111(11, 0, 0) → 3
tt2() → 12
activate2(0) → 13
activate2(0) → 14
U122(12, 13, 14) → 3
activate2(7) → 16
activate2(6) → 17
plus2(16, 17) → 15
s2(15) → 1
U111(11, 0, 9) → 8
activate3(14) → 19
activate3(13) → 20
plus3(19, 20) → 18
s3(18) → 3
activate2(9) → 14
U122(12, 13, 14) → 8
U111(11, 0, 16) → 15
activate2(16) → 14
U122(12, 13, 14) → 15
s3(18) → 8
U111(11, 0, 19) → 18
activate2(19) → 14
U122(12, 13, 14) → 18
s3(18) → 15
s3(18) → 18
0 → 3
0 → 4
0 → 6
0 → 7
0 → 9
0 → 10
0 → 13
0 → 14
9 → 8
9 → 14
7 → 16
6 → 17
16 → 15
16 → 14
14 → 19
13 → 20
19 → 18
19 → 14