WORST_CASE(?,O(n^1))
proof of input_FO0EE39H7t.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 38 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:

   plus#2(0, x12) -> x12
   plus#2(S(x4), x2) -> S(plus#2(x4, x2))
   fold#3(Nil) -> 0
   fold#3(Cons(x4, x2)) -> plus#2(x4, fold#3(x2))
   main(x1) -> fold#3(x1)

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:

   plus#2(0, x12) -> x12
   plus#2(S(x4), x2) -> S(plus#2(x4, x2))
   fold#3(Nil) -> 0
   fold#3(Cons(x4, x2)) -> plus#2(x4, fold#3(x2))
   main(x1) -> fold#3(x1)

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, 3]
transitions: 
00() -> 0
S0(0) -> 0
Nil0() -> 0
Cons0(0, 0) -> 0
plus#20(0, 0) -> 1
fold#30(0) -> 2
main0(0) -> 3
plus#21(0, 0) -> 4
S1(4) -> 1
01() -> 2
fold#31(0) -> 5
plus#21(0, 5) -> 2
fold#31(0) -> 3
plus#21(0, 5) -> 4
S1(4) -> 2
S1(4) -> 4
01() -> 3
01() -> 5
plus#21(0, 5) -> 3
plus#21(0, 5) -> 5
S1(4) -> 3
S1(4) -> 5
0 -> 1
0 -> 4
5 -> 2
5 -> 3
5 -> 4

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(4)
BOUNDS(1, n^1)