WORST_CASE(?,O(n^1))
proof of input_ufv7G9K9aL.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 0 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(s(X)) -> f(X)
   g(cons(0, Y)) -> g(Y)
   g(cons(s(X), Y)) -> s(X)
   h(cons(X, Y)) -> h(g(cons(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(s(X)) -> f(X)
   g(cons(0, Y)) -> g(Y)
   g(cons(s(X), Y)) -> s(X)
   h(cons(X, Y)) -> h(g(cons(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 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3]
transitions: 
s0(0) -> 0
cons0(0, 0) -> 0
00() -> 0
f0(0) -> 1
g0(0) -> 2
h0(0) -> 3
f1(0) -> 1
g1(0) -> 2
s1(0) -> 2
cons1(0, 0) -> 5
g1(5) -> 4
h1(4) -> 3
g1(0) -> 4
s1(0) -> 4

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(4)
BOUNDS(1, n^1)