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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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

   odd(Cons(x, xs)) -> even(xs)
   odd(Nil) -> False
   even(Cons(x, xs)) -> odd(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   even(Nil) -> True
   evenodd(x) -> even(x)

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:

   odd(Cons(x, xs)) -> even(xs)
   odd(Nil) -> False
   even(Cons(x, xs)) -> odd(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   even(Nil) -> True
   evenodd(x) -> even(x)

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, 4]
transitions: 
Cons0(0, 0) -> 0
Nil0() -> 0
False0() -> 0
True0() -> 0
odd0(0) -> 1
even0(0) -> 2
notEmpty0(0) -> 3
evenodd0(0) -> 4
even1(0) -> 1
False1() -> 1
odd1(0) -> 2
True1() -> 3
False1() -> 3
True1() -> 2
even1(0) -> 4
even1(0) -> 2
False1() -> 2
odd1(0) -> 1
odd1(0) -> 4
True1() -> 1
True1() -> 4
False1() -> 4

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(4)
BOUNDS(1, n^1)