WORST_CASE(?,O(n^1))
proof of input_z2U9Lff0CR.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 0 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:

   compS_f#1(compS_f(x2), x1) -> compS_f#1(x2, S(x1))
   compS_f#1(id, x3) -> S(x3)
   iter#3(0) -> id
   iter#3(S(x6)) -> compS_f(iter#3(x6))
   main(0) -> 0
   main(S(x9)) -> compS_f#1(iter#3(x9), 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:

   compS_f#1(compS_f(x2), x1) -> compS_f#1(x2, S(x1))
   compS_f#1(id, x3) -> S(x3)
   iter#3(0) -> id
   iter#3(S(x6)) -> compS_f(iter#3(x6))
   main(0) -> 0
   main(S(x9)) -> compS_f#1(iter#3(x9), 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 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3]
transitions: 
compS_f0(0) -> 0
S0(0) -> 0
id0() -> 0
00() -> 0
compS_f#10(0, 0) -> 1
iter#30(0) -> 2
main0(0) -> 3
S1(0) -> 4
compS_f#11(0, 4) -> 1
S1(0) -> 1
id1() -> 2
iter#31(0) -> 5
compS_f1(5) -> 2
01() -> 3
iter#31(0) -> 6
01() -> 7
compS_f#11(6, 7) -> 3
S1(4) -> 4
S1(4) -> 1
id1() -> 5
id1() -> 6
compS_f1(5) -> 5
compS_f1(5) -> 6
S2(7) -> 8
compS_f#12(5, 8) -> 3
S2(7) -> 3
S2(8) -> 8
S2(8) -> 3

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(4)
BOUNDS(1, n^1)