WORST_CASE(?,O(n^1))
proof of input_lNZcJfndbZ.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 53 ms]
(4) BOUNDS(1, n^1)


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(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:

   max(L(x)) -> x
   max(N(L(0), L(y))) -> y
   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
   max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(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:

   max(L(x)) -> x
   max(N(L(0), L(y))) -> y
   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
   max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(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 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1]
transitions: 
L0(0) -> 0
N0(0, 0) -> 0
00() -> 0
s0(0) -> 0
max0(0) -> 1
L1(0) -> 4
L1(0) -> 5
N1(4, 5) -> 3
max1(3) -> 2
s1(2) -> 1
N1(0, 0) -> 8
max1(8) -> 7
L1(7) -> 6
N1(4, 6) -> 3
max1(3) -> 1
s1(2) -> 7
max1(3) -> 7
L1(2) -> 5
0 -> 1
0 -> 7
0 -> 2
7 -> 1
7 -> 2
2 -> 1
2 -> 7

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(4)
BOUNDS(1, n^1)