WORST_CASE(?,O(n^1))
proof of input_Y5WZmne4pp.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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:

   addlist(Cons(x, xs'), Cons(S(0), xs)) -> Cons(S(x), addlist(xs', xs))
   addlist(Cons(S(0), xs'), Cons(x, xs)) -> Cons(S(x), addlist(xs', xs))
   addlist(Nil, ys) -> Nil
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs, ys) -> addlist(xs, ys)

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:

   addlist(Cons(x, xs'), Cons(S(0), xs)) -> Cons(S(x), addlist(xs', xs))
   addlist(Cons(S(0), xs'), Cons(x, xs)) -> Cons(S(x), addlist(xs', xs))
   addlist(Nil, ys) -> Nil
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs, ys) -> addlist(xs, ys)

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: 
Cons0(0, 0) -> 0
S0(0) -> 0
00() -> 0
Nil0() -> 0
True0() -> 0
False0() -> 0
addlist0(0, 0) -> 1
notEmpty0(0) -> 2
goal0(0, 0) -> 3
S1(0) -> 4
addlist1(0, 0) -> 5
Cons1(4, 5) -> 1
Nil1() -> 1
True1() -> 2
False1() -> 2
addlist1(0, 0) -> 3
Cons1(4, 5) -> 3
Cons1(4, 5) -> 5
Nil1() -> 3
Nil1() -> 5

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(4)
BOUNDS(1, n^1)