WORST_CASE(?,O(n^1))
proof of input_N7kFNhDwYJ.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 44 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:

   dec(Cons(Nil, Nil)) -> Nil
   dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs))
   dec(Cons(Cons(x, xs), Nil)) -> dec(Nil)
   dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs))
   isNilNil(Cons(Nil, Nil)) -> True
   isNilNil(Cons(Nil, Cons(x, xs))) -> False
   isNilNil(Cons(Cons(x, xs), Nil)) -> False
   isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False
   nestdec(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)))))))))))))))))
   nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs)))
   number17(n) -> 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) -> nestdec(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:

   dec(Cons(Nil, Nil)) -> Nil
   dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs))
   dec(Cons(Cons(x, xs), Nil)) -> dec(Nil)
   dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs))
   isNilNil(Cons(Nil, Nil)) -> True
   isNilNil(Cons(Nil, Cons(x, xs))) -> False
   isNilNil(Cons(Cons(x, xs), Nil)) -> False
   isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False
   nestdec(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)))))))))))))))))
   nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs)))
   number17(n) -> 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) -> nestdec(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, 4, 5]
transitions: 
Cons0(0, 0) -> 0
Nil0() -> 0
True0() -> 0
False0() -> 0
dec0(0) -> 1
isNilNil0(0) -> 2
nestdec0(0) -> 3
number170(0) -> 4
goal0(0) -> 5
Nil1() -> 1
Cons1(0, 0) -> 6
dec1(6) -> 1
Nil1() -> 7
dec1(7) -> 1
True1() -> 2
False1() -> 2
Nil1() -> 8
Nil1() -> 11
Cons1(8, 11) -> 10
Cons1(8, 10) -> 9
Cons1(8, 9) -> 9
Cons1(8, 9) -> 3
dec1(6) -> 12
nestdec1(12) -> 3
Cons1(8, 9) -> 4
nestdec1(0) -> 5
Nil1() -> 12
dec1(7) -> 12
Cons1(8, 9) -> 5
nestdec1(12) -> 5
Nil2() -> 13
Nil2() -> 16
Cons2(13, 16) -> 15
Cons2(13, 15) -> 14
Cons2(13, 14) -> 14
Cons2(13, 14) -> 3
Cons2(13, 14) -> 5

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(4)
BOUNDS(1, n^1)