WORST_CASE(?,O(n^1))
proof of input_SckCMh3yzh.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 57 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:

   revApp#2(Nil, x16) -> x16
   revApp#2(Cons(x6, x4), x2) -> revApp#2(x4, Cons(x6, x2))
   dfsAcc#3(Leaf(x8), x16) -> Cons(x8, x16)
   dfsAcc#3(Node(x6, x4), x2) -> dfsAcc#3(x4, dfsAcc#3(x6, x2))
   main(x1) -> revApp#2(dfsAcc#3(x1, Nil), Nil)

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:

   revApp#2(Nil, x16) -> x16
   revApp#2(Cons(x6, x4), x2) -> revApp#2(x4, Cons(x6, x2))
   dfsAcc#3(Leaf(x8), x16) -> Cons(x8, x16)
   dfsAcc#3(Node(x6, x4), x2) -> dfsAcc#3(x4, dfsAcc#3(x6, x2))
   main(x1) -> revApp#2(dfsAcc#3(x1, Nil), Nil)

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: 
Nil0() -> 0
Cons0(0, 0) -> 0
Leaf0(0) -> 0
Node0(0, 0) -> 0
revApp#20(0, 0) -> 1
dfsAcc#30(0, 0) -> 2
main0(0) -> 3
Cons1(0, 0) -> 4
revApp#21(0, 4) -> 1
Cons1(0, 0) -> 2
dfsAcc#31(0, 0) -> 5
dfsAcc#31(0, 5) -> 2
Nil1() -> 7
dfsAcc#31(0, 7) -> 6
Nil1() -> 8
revApp#21(6, 8) -> 3
Cons1(0, 4) -> 4
Cons1(0, 5) -> 2
Cons1(0, 0) -> 5
Cons1(0, 7) -> 6
dfsAcc#31(0, 5) -> 5
dfsAcc#31(0, 7) -> 5
dfsAcc#31(0, 5) -> 6
Cons2(0, 8) -> 9
revApp#22(7, 9) -> 3
Cons1(0, 5) -> 5
Cons1(0, 7) -> 5
Cons1(0, 5) -> 6
revApp#22(5, 9) -> 3
Cons2(0, 9) -> 9
revApp#22(0, 9) -> 3
Cons1(0, 9) -> 4
revApp#21(0, 4) -> 3
0 -> 1
4 -> 1
4 -> 3
9 -> 3

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(4)
BOUNDS(1, n^1)