WORST_CASE(?,O(n^1))
proof of input_Ni6z9HKaG6.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 44 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:

   dbl(S(0), S(0)) -> S(S(S(S(0))))
   save(S(x)) -> dbl(0, save(x))
   save(0) -> 0
   dbl(0, y) -> y

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:

   dbl(S(0), S(0)) -> S(S(S(S(0))))
   save(S(x)) -> dbl(0, save(x))
   save(0) -> 0
   dbl(0, y) -> y

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, 2]
transitions: 
S0(0) -> 0
00() -> 0
dbl0(0, 0) -> 1
save0(0) -> 2
01() -> 5
S1(5) -> 4
S1(4) -> 3
S1(3) -> 3
S1(3) -> 1
01() -> 6
save1(0) -> 7
dbl1(6, 7) -> 2
01() -> 2
dbl1(6, 7) -> 7
01() -> 7
0 -> 1
7 -> 2

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(4)
BOUNDS(1, n^1)