3.59/1.89	WORST_CASE(Omega(n^1), O(n^1))
3.59/1.89	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.59/1.89	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.59/1.89	
3.59/1.89	
3.59/1.89	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0)).
3.59/1.89	
3.59/1.89	(0) CpxIntTrs
3.59/1.89	(1) Koat2 Proof [FINISHED, 16 ms]
3.59/1.89	(2) BOUNDS(1, max(1, 1 + Arg_0))
3.59/1.89	(3) Loat Proof [FINISHED, 206 ms]
3.59/1.89	(4) BOUNDS(n^1, INF)
3.59/1.89	
3.59/1.89	
3.59/1.89	----------------------------------------
3.59/1.89	
3.59/1.89	(0)
3.59/1.89	Obligation:
3.59/1.89	Complexity Int TRS consisting of the following rules:
3.59/1.89	f0(A) -> Com_1(f1(A)) :|: TRUE
3.59/1.89	f1(A) -> Com_1(f1(A - 1000)) :|: A >= 1001
3.59/1.89	
3.59/1.89	The start-symbols are:[f0_1]
3.59/1.89	
3.59/1.89	
3.59/1.89	----------------------------------------
3.59/1.89	
3.59/1.89	(1) Koat2 Proof (FINISHED)
3.59/1.89	YES( ?, max([1, 1+Arg_0]) {O(n)})
3.59/1.89	
3.59/1.89	
3.59/1.89	
3.59/1.89	Initial Complexity Problem:
3.59/1.89	
3.59/1.89	  Start:  f0  
3.59/1.89	
3.59/1.89	  Program_Vars:  Arg_0  
3.59/1.89	
3.59/1.89	  Temp_Vars:    
3.59/1.89	
3.59/1.89	  Locations:  f0, f1  
3.59/1.89	
3.59/1.89	  Transitions:
3.59/1.89	
3.59/1.89	  f0(Arg_0) -> f1(Arg_0):|:
3.59/1.89	
3.59/1.89	  f1(Arg_0) -> f1(Arg_0-1000):|:1001 <= Arg_0  
3.59/1.89	
3.59/1.89	
3.59/1.89	
3.59/1.89	Timebounds: 
3.59/1.89	
3.59/1.89	  Overall timebound: max([1, 1+Arg_0]) {O(n)}
3.59/1.89	
3.59/1.89	  0: f0->f1: 1 {O(1)}
3.59/1.89	
3.59/1.89	  1: f1->f1: max([0, Arg_0]) {O(n)}
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	Costbounds:
3.59/1.90	
3.59/1.90	  Overall costbound: max([1, 1+Arg_0]) {O(n)}
3.59/1.90	
3.59/1.90	  0: f0->f1: 1 {O(1)}
3.59/1.90	
3.59/1.90	  1: f1->f1: max([0, Arg_0]) {O(n)}
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	Sizebounds:
3.59/1.90	
3.59/1.90	`Lower:
3.59/1.90	
3.59/1.90	  0: f0->f1, Arg_0: Arg_0 {O(n)}
3.59/1.90	
3.59/1.90	  1: f1->f1, Arg_0: 1 {O(1)}
3.59/1.90	
3.59/1.90	`Upper:
3.59/1.90	
3.59/1.90	  0: f0->f1, Arg_0: Arg_0 {O(n)}
3.59/1.90	
3.59/1.90	  1: f1->f1, Arg_0: Arg_0 {O(n)}
3.59/1.90	
3.59/1.90	
3.59/1.90	----------------------------------------
3.59/1.90	
3.59/1.90	(2)
3.59/1.90	BOUNDS(1, max(1, 1 + Arg_0))
3.59/1.90	
3.59/1.90	----------------------------------------
3.59/1.90	
3.59/1.90	(3) Loat Proof (FINISHED)
3.59/1.90	
3.59/1.90	
3.59/1.90	### Pre-processing the ITS problem ###
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	Initial linear ITS problem
3.59/1.90	
3.59/1.90	   Start location: f0
3.59/1.90	
3.59/1.90	      0: f0 -> f1 : [], cost: 1
3.59/1.90	
3.59/1.90	      1: f1 -> f1 : A'=-1000+A, [ A>=1001 ], cost: 1
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	### Simplification by acceleration and chaining ###
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	Accelerating simple loops of location 1.
3.59/1.90	
3.59/1.90	   Accelerating the following rules:
3.59/1.90	
3.59/1.90	      1: f1 -> f1 : A'=-1000+A, [ A>=1001 ], cost: 1
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	   Accelerated rule 1 with metering function meter (where 1000*meter==-1000+A), yielding the new rule 2.
3.59/1.90	
3.59/1.90	   Removing the simple loops: 1.
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	Accelerated all simple loops using metering functions (where possible):
3.59/1.90	
3.59/1.90	   Start location: f0
3.59/1.90	
3.59/1.90	      0: f0 -> f1 : [], cost: 1
3.59/1.90	
3.59/1.90	      2: f1 -> f1 : A'=-1000*meter+A, [ A>=1001 && 1000*meter==-1000+A && meter>=1 ], cost: meter
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	Chained accelerated rules (with incoming rules):
3.59/1.90	
3.59/1.90	   Start location: f0
3.59/1.90	
3.59/1.90	      0: f0 -> f1 : [], cost: 1
3.59/1.90	
3.59/1.90	      3: f0 -> f1 : A'=-1000*meter+A, [ A>=1001 && 1000*meter==-1000+A && meter>=1 ], cost: 1+meter
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	Removed unreachable locations (and leaf rules with constant cost):
3.59/1.90	
3.59/1.90	   Start location: f0
3.59/1.90	
3.59/1.90	      3: f0 -> f1 : A'=-1000*meter+A, [ A>=1001 && 1000*meter==-1000+A && meter>=1 ], cost: 1+meter
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	### Computing asymptotic complexity ###
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	Fully simplified ITS problem
3.59/1.90	
3.59/1.90	   Start location: f0
3.59/1.90	
3.59/1.90	      3: f0 -> f1 : A'=-1000*meter+A, [ A>=1001 && 1000*meter==-1000+A && meter>=1 ], cost: 1+meter
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	Computing asymptotic complexity for rule 3
3.59/1.90	
3.59/1.90	   Solved the limit problem by the following transformations:
3.59/1.90	
3.59/1.90	      Created initial limit problem:
3.59/1.90	
3.59/1.90	      1+meter (+), -1000+A (+/+!), 1001+1000*meter-A (+/+!), -999-1000*meter+A (+/+!) [not solved]
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	      applying transformation rule (C) using substitution {A==1000+1000*meter}
3.59/1.90	
3.59/1.90	      resulting limit problem:
3.59/1.90	
3.59/1.90	      1 (+/+!), 1+meter (+), 1000*meter (+/+!) [not solved]
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	      applying transformation rule (B), deleting 1 (+/+!)
3.59/1.90	
3.59/1.90	      resulting limit problem:
3.59/1.90	
3.59/1.90	      1+meter (+), 1000*meter (+/+!) [not solved]
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	      removing all constraints (solved by SMT)
3.59/1.90	
3.59/1.90	      resulting limit problem: [solved]
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	      applying transformation rule (C) using substitution {meter==n}
3.59/1.90	
3.59/1.90	      resulting limit problem:
3.59/1.90	
3.59/1.90	      [solved]
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	   Solved the limit problem by the following transformations:
3.59/1.90	
3.59/1.90	      Created initial limit problem:
3.59/1.90	
3.59/1.90	      1+meter (+), -1000+A (+/+!), 1001+1000*meter-A (+/+!), -999-1000*meter+A (+/+!) [not solved]
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	      applying transformation rule (C) using substitution {A==1000+1000*meter}
3.59/1.90	
3.59/1.90	      resulting limit problem:
3.59/1.90	
3.59/1.90	      1 (+/+!), 1+meter (+), 1000*meter (+/+!) [not solved]
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	      applying transformation rule (B), deleting 1 (+/+!)
3.59/1.90	
3.59/1.90	      resulting limit problem:
3.59/1.90	
3.59/1.90	      1+meter (+), 1000*meter (+/+!) [not solved]
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	      removing all constraints (solved by SMT)
3.59/1.90	
3.59/1.90	      resulting limit problem: [solved]
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	      applying transformation rule (C) using substitution {meter==n}
3.59/1.90	
3.59/1.90	      resulting limit problem:
3.59/1.90	
3.59/1.90	      [solved]
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	   Solution:
3.59/1.90	
3.59/1.90	      meter / n
3.59/1.90	
3.59/1.90	      A / 1000+1000*n
3.59/1.90	
3.59/1.90	   Resulting cost 1+n has complexity: Poly(n^1)
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	Found new complexity Poly(n^1).
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	Obtained the following overall complexity (w.r.t. the length of the input n):
3.59/1.90	
3.59/1.90	   Complexity:  Poly(n^1)
3.59/1.90	
3.59/1.90	   Cpx degree:  1
3.59/1.90	
3.59/1.90	   Solved cost: 1+n
3.59/1.90	
3.59/1.90	   Rule cost:   1+meter
3.59/1.90	
3.59/1.90	   Rule guard:  [ A>=1001 && 1000*meter==-1000+A ]
3.59/1.90	
3.59/1.90	
3.59/1.90	
3.59/1.90	WORST_CASE(Omega(n^1),?)
3.59/1.90	
3.59/1.90	
3.59/1.90	----------------------------------------
3.59/1.90	
3.59/1.90	(4)
3.59/1.90	BOUNDS(n^1, INF)
3.84/1.92	EOF