WORST_CASE(?,O(n^1))
proof of input_PkNguFxpvX.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 13 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(x, y) -> g(x, y)
   g(h(x), y) -> h(f(x, y))
   g(h(x), y) -> h(g(x, 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(x, y) -> g(x, y)
   g(h(x), y) -> h(f(x, y))
   g(h(x), y) -> h(g(x, 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, 2]
transitions: 
h0(0) -> 0
f0(0, 0) -> 1
g0(0, 0) -> 2
g1(0, 0) -> 1
f1(0, 0) -> 3
h1(3) -> 2
g1(0, 0) -> 4
h1(4) -> 2
g2(0, 0) -> 3
h1(3) -> 1
h1(3) -> 4
h1(4) -> 1
h1(4) -> 4
h1(3) -> 3
h1(4) -> 3

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(4)
BOUNDS(1, n^1)