WORST_CASE(?,O(n^1))
proof of input_0tH42SeaVd.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 22 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:

   merge(x, nil) -> x
   merge(nil, y) -> y
   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

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:

   merge(x, nil) -> x
   merge(nil, y) -> y
   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

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: 
nil0() -> 0
++0(0, 0) -> 0
u0() -> 0
v0() -> 0
merge0(0, 0) -> 1
u1() -> 4
v1() -> 5
++1(4, 5) -> 3
merge1(0, 3) -> 2
++1(0, 2) -> 1
++1(0, 0) -> 7
v1() -> 8
merge1(7, 8) -> 6
++1(4, 6) -> 1
++1(0, 2) -> 2
++1(4, 6) -> 2
0 -> 1
3 -> 2

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(4)
BOUNDS(1, n^1)