WORST_CASE(?,O(n^1))
proof of input_F2mqry8XS9.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 35 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(nil) -> nil
   f(.(nil, y)) -> .(nil, f(y))
   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
   g(nil) -> nil
   g(.(x, nil)) -> .(g(x), nil)
   g(.(x, .(y, z))) -> g(.(.(x, y), 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:

   f(nil) -> nil
   f(.(nil, y)) -> .(nil, f(y))
   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
   g(nil) -> nil
   g(.(x, nil)) -> .(g(x), nil)
   g(.(x, .(y, z))) -> g(.(.(x, y), 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 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2]
transitions: 
nil0() -> 0
.0(0, 0) -> 0
f0(0) -> 1
g0(0) -> 2
nil1() -> 1
nil1() -> 3
f1(0) -> 4
.1(3, 4) -> 1
.1(0, 0) -> 6
.1(0, 6) -> 5
f1(5) -> 1
nil1() -> 2
g1(0) -> 7
nil1() -> 8
.1(7, 8) -> 2
.1(0, 0) -> 10
.1(10, 0) -> 9
g1(9) -> 2
nil1() -> 4
.1(3, 4) -> 4
f1(6) -> 4
f1(5) -> 4
.1(0, 6) -> 6
nil1() -> 7
.1(7, 8) -> 7
g1(10) -> 7
g1(9) -> 7
.1(10, 0) -> 10

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(4)
BOUNDS(1, n^1)