WORST_CASE(?,O(n^1))
proof of input_4TzYBaYshu.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 14 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:

   f(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(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:

   f(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(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 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1]
transitions: 
00() -> 0
s0(0) -> 0
*0(0, 0) -> 0
+0(0, 0) -> 0
f0(0) -> 1
01() -> 2
s1(2) -> 1
s1(2) -> 3
s1(3) -> 1
s1(3) -> 4
01() -> 6
f1(6) -> 5
*1(4, 5) -> 1
s1(3) -> 7
f1(0) -> 8
+1(7, 8) -> 1
f1(0) -> 9
f1(0) -> 10
*1(9, 10) -> 1
s1(2) -> 8
s1(2) -> 9
s1(2) -> 10
02() -> 11
s2(11) -> 5
s1(3) -> 8
s1(3) -> 9
s1(3) -> 10
*1(4, 5) -> 8
*1(4, 5) -> 9
*1(4, 5) -> 10
+1(7, 8) -> 8
+1(7, 8) -> 9
+1(7, 8) -> 10
*1(9, 10) -> 8
*1(9, 10) -> 9
*1(9, 10) -> 10

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(4)
BOUNDS(1, n^1)