WORST_CASE(?,O(n^1))
proof of input_xmVhRgnWuN.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 38 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:

   decrease(Cons(x, xs)) -> decrease(xs)
   decrease(Nil) -> number42(Nil)
   number42(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
   goal(x) -> decrease(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:

   decrease(Cons(x, xs)) -> decrease(xs)
   decrease(Nil) -> number42(Nil)
   number42(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
   goal(x) -> decrease(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 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: 
Cons0(0, 0) -> 0
Nil0() -> 0
decrease0(0) -> 1
number420(0) -> 2
goal0(0) -> 3
decrease1(0) -> 1
Nil1() -> 4
number421(4) -> 1
Nil1() -> 5
Nil1() -> 8
Cons1(5, 8) -> 7
Cons1(5, 7) -> 6
Cons1(5, 6) -> 6
Cons1(5, 6) -> 2
decrease1(0) -> 3
number421(4) -> 3
Nil2() -> 9
Nil2() -> 12
Cons2(9, 12) -> 11
Cons2(9, 11) -> 10
Cons2(9, 10) -> 10
Cons2(9, 10) -> 1
Cons2(9, 10) -> 3

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(4)
BOUNDS(1, n^1)