WORST_CASE(?,O(n^1))
proof of input_2JKwJmADqS.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 68 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:

   +(0, y) -> y
   +(s(x), 0) -> s(x)
   +(s(x), s(y)) -> s(+(s(x), +(y, 0)))

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:

   +(0, y) -> y
   +(s(x), 0) -> s(x)
   +(s(x), s(y)) -> s(+(s(x), +(y, 0)))

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 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

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(4)
BOUNDS(1, n^1)