WORST_CASE(?,O(n^1))
proof of input_hDl7xbwJil.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 229 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:

   h(x, c(y, z)) -> h(c(s(y), x), z)
   h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z)))

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:

   h(x, c(y, z)) -> h(c(s(y), x), z)
   h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z)))

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: 
c0(0, 0) -> 0
s0(0) -> 0
00() -> 0
h0(0, 0) -> 1
s1(0) -> 3
c1(3, 0) -> 2
h1(2, 0) -> 1
01() -> 6
s1(6) -> 5
c1(0, 0) -> 7
c1(5, 7) -> 4
h1(0, 4) -> 1
c1(3, 2) -> 2
s2(5) -> 9
c2(9, 0) -> 8
h2(8, 7) -> 1
c1(0, 4) -> 7
h1(2, 4) -> 1
s2(0) -> 9
c2(9, 8) -> 8
h2(8, 0) -> 1
h2(8, 4) -> 1
c1(5, 7) -> 7
c2(9, 2) -> 8
c1(3, 8) -> 2
c1(0, 7) -> 7
c1(5, 0) -> 7
c1(5, 4) -> 7
h1(8, 4) -> 1

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(4)
BOUNDS(1, n^1)