WORST_CASE(?,O(n^1))
proof of input_HD5HvCrEnQ.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 30 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:

   add0(x', Cons(x, xs)) -> add0(Cons(Cons(Nil, Nil), x'), xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   add0(x, Nil) -> x
   goal(x, y) -> add0(x, 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:

   add0(x', Cons(x, xs)) -> add0(Cons(Cons(Nil, Nil), x'), xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   add0(x, Nil) -> x
   goal(x, y) -> add0(x, 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, 3]
transitions: 
Cons0(0, 0) -> 0
Nil0() -> 0
True0() -> 0
False0() -> 0
add00(0, 0) -> 1
notEmpty0(0) -> 2
goal0(0, 0) -> 3
Nil1() -> 6
Nil1() -> 7
Cons1(6, 7) -> 5
Cons1(5, 0) -> 4
add01(4, 0) -> 1
True1() -> 2
False1() -> 2
add01(0, 0) -> 3
Cons1(5, 4) -> 4
add01(4, 0) -> 3
0 -> 1
0 -> 3
4 -> 1
4 -> 3

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(4)
BOUNDS(1, n^1)