WORST_CASE(?,O(n^1))
proof of input_KWe1ALO0Lf.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
(3) CpxTrsMatchBoundsTAProof [FINISHED, 51 ms]
(4) BOUNDS(1, n^1)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


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)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


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)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) CpxTrsMatchBoundsTAProof (FINISHED)
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

----------------------------------------

(4)
BOUNDS(1, n^1)