3.95/1.78	WORST_CASE(Omega(n^1), O(n^1))
3.95/1.79	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
3.95/1.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.95/1.79	
3.95/1.79	
3.95/1.79	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 2 + Arg_0)).
3.95/1.79	
3.95/1.79	(0) CpxIntTrs
3.95/1.79	(1) Koat2 Proof [FINISHED, 26 ms]
3.95/1.79	(2) BOUNDS(1, max(1, 2 + Arg_0))
3.95/1.79	(3) Loat Proof [FINISHED, 129 ms]
3.95/1.79	(4) BOUNDS(n^1, INF)
3.95/1.79	
3.95/1.79	
3.95/1.79	----------------------------------------
3.95/1.79	
3.95/1.79	(0)
3.95/1.79	Obligation:
3.95/1.79	Complexity Int TRS consisting of the following rules:
3.95/1.79	f1(A) -> Com_1(f0(A)) :|: TRUE
3.95/1.79	f0(A) -> Com_1(f0(A - 1)) :|: A >= 0
3.95/1.79	
3.95/1.79	The start-symbols are:[f1_1]
3.95/1.79	
3.95/1.79	
3.95/1.79	----------------------------------------
3.95/1.79	
3.95/1.79	(1) Koat2 Proof (FINISHED)
3.95/1.79	YES( ?, max([1, 2+Arg_0]) {O(n)})
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	Initial Complexity Problem:
3.95/1.79	
3.95/1.79	  Start:  f1  
3.95/1.79	
3.95/1.79	  Program_Vars:  Arg_0  
3.95/1.79	
3.95/1.79	  Temp_Vars:    
3.95/1.79	
3.95/1.79	  Locations:  f0, f1  
3.95/1.79	
3.95/1.79	  Transitions:
3.95/1.79	
3.95/1.79	  f0(Arg_0) -> f0(Arg_0-1):|:0 <= Arg_0
3.95/1.79	
3.95/1.79	  f1(Arg_0) -> f0(Arg_0):|:  
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	Timebounds: 
3.95/1.79	
3.95/1.79	  Overall timebound: max([1, 2+Arg_0]) {O(n)}
3.95/1.79	
3.95/1.79	  1: f0->f0: max([0, 1+Arg_0]) {O(n)}
3.95/1.79	
3.95/1.79	  0: f1->f0: 1 {O(1)}
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	Costbounds:
3.95/1.79	
3.95/1.79	  Overall costbound: max([1, 2+Arg_0]) {O(n)}
3.95/1.79	
3.95/1.79	  1: f0->f0: max([0, 1+Arg_0]) {O(n)}
3.95/1.79	
3.95/1.79	  0: f1->f0: 1 {O(1)}
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	Sizebounds:
3.95/1.79	
3.95/1.79	`Lower:
3.95/1.79	
3.95/1.79	  1: f0->f0, Arg_0: -1 {O(1)}
3.95/1.79	
3.95/1.79	  0: f1->f0, Arg_0: Arg_0 {O(n)}
3.95/1.79	
3.95/1.79	`Upper:
3.95/1.79	
3.95/1.79	  1: f0->f0, Arg_0: Arg_0 {O(n)}
3.95/1.79	
3.95/1.79	  0: f1->f0, Arg_0: Arg_0 {O(n)}
3.95/1.79	
3.95/1.79	
3.95/1.79	----------------------------------------
3.95/1.79	
3.95/1.79	(2)
3.95/1.79	BOUNDS(1, max(1, 2 + Arg_0))
3.95/1.79	
3.95/1.79	----------------------------------------
3.95/1.79	
3.95/1.79	(3) Loat Proof (FINISHED)
3.95/1.79	
3.95/1.79	
3.95/1.79	### Pre-processing the ITS problem ###
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	Initial linear ITS problem
3.95/1.79	
3.95/1.79	   Start location: f1
3.95/1.79	
3.95/1.79	      0: f1 -> f0 : [], cost: 1
3.95/1.79	
3.95/1.79	      1: f0 -> f0 : A'=-1+A, [ A>=0 ], cost: 1
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	### Simplification by acceleration and chaining ###
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	Accelerating simple loops of location 1.
3.95/1.79	
3.95/1.79	   Accelerating the following rules:
3.95/1.79	
3.95/1.79	      1: f0 -> f0 : A'=-1+A, [ A>=0 ], cost: 1
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	   Accelerated rule 1 with metering function 1+A, yielding the new rule 2.
3.95/1.79	
3.95/1.79	   Removing the simple loops: 1.
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	Accelerated all simple loops using metering functions (where possible):
3.95/1.79	
3.95/1.79	   Start location: f1
3.95/1.79	
3.95/1.79	      0: f1 -> f0 : [], cost: 1
3.95/1.79	
3.95/1.79	      2: f0 -> f0 : A'=-1, [ A>=0 ], cost: 1+A
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	Chained accelerated rules (with incoming rules):
3.95/1.79	
3.95/1.79	   Start location: f1
3.95/1.79	
3.95/1.79	      0: f1 -> f0 : [], cost: 1
3.95/1.79	
3.95/1.79	      3: f1 -> f0 : A'=-1, [ A>=0 ], cost: 2+A
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	Removed unreachable locations (and leaf rules with constant cost):
3.95/1.79	
3.95/1.79	   Start location: f1
3.95/1.79	
3.95/1.79	      3: f1 -> f0 : A'=-1, [ A>=0 ], cost: 2+A
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	### Computing asymptotic complexity ###
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	Fully simplified ITS problem
3.95/1.79	
3.95/1.79	   Start location: f1
3.95/1.79	
3.95/1.79	      3: f1 -> f0 : A'=-1, [ A>=0 ], cost: 2+A
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	Computing asymptotic complexity for rule 3
3.95/1.79	
3.95/1.79	   Solved the limit problem by the following transformations:
3.95/1.79	
3.95/1.79	      Created initial limit problem:
3.95/1.79	
3.95/1.79	      2+A (+), 1+A (+/+!) [not solved]
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	      removing all constraints (solved by SMT)
3.95/1.79	
3.95/1.79	      resulting limit problem: [solved]
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	      applying transformation rule (C) using substitution {A==n}
3.95/1.79	
3.95/1.79	      resulting limit problem:
3.95/1.79	
3.95/1.79	      [solved]
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	   Solution:
3.95/1.79	
3.95/1.79	      A / n
3.95/1.79	
3.95/1.79	   Resulting cost 2+n has complexity: Poly(n^1)
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	Found new complexity Poly(n^1).
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	Obtained the following overall complexity (w.r.t. the length of the input n):
3.95/1.79	
3.95/1.79	   Complexity:  Poly(n^1)
3.95/1.79	
3.95/1.79	   Cpx degree:  1
3.95/1.79	
3.95/1.79	   Solved cost: 2+n
3.95/1.79	
3.95/1.79	   Rule cost:   2+A
3.95/1.79	
3.95/1.79	   Rule guard:  [ A>=0 ]
3.95/1.79	
3.95/1.79	
3.95/1.79	
3.95/1.79	WORST_CASE(Omega(n^1),?)
3.95/1.79	
3.95/1.79	
3.95/1.79	----------------------------------------
3.95/1.79	
3.95/1.79	(4)
3.95/1.79	BOUNDS(n^1, INF)
3.95/1.80	EOF