WORST_CASE(?,O(n^1))
proof of input_961vkkxPQu.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 30 ms]
(4) BOUNDS(1, n^1)


----------------------------------------

(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:

   anchored(Cons(x, xs), y) -> anchored(xs, Cons(Cons(Nil, Nil), y))
   anchored(Nil, y) -> y
   goal(x, y) -> anchored(x, y)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(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:

   anchored(Cons(x, xs), y) -> anchored(xs, Cons(Cons(Nil, Nil), y))
   anchored(Nil, y) -> y
   goal(x, y) -> anchored(x, y)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(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: 
Cons0(0, 0) -> 0
Nil0() -> 0
anchored0(0, 0) -> 1
goal0(0, 0) -> 2
Nil1() -> 5
Nil1() -> 6
Cons1(5, 6) -> 4
Cons1(4, 0) -> 3
anchored1(0, 3) -> 1
anchored1(0, 0) -> 2
Cons1(4, 3) -> 3
anchored1(0, 3) -> 2
0 -> 1
0 -> 2
3 -> 1
3 -> 2

----------------------------------------

(4)
BOUNDS(1, n^1)