3.47/1.78	WORST_CASE(Omega(n^1), O(n^1))
3.47/1.79	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.47/1.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.47/1.79	
3.47/1.79	
3.47/1.79	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(2, 3 + Arg_0)).
3.47/1.79	
3.47/1.79	(0) CpxIntTrs
3.47/1.79	(1) Koat2 Proof [FINISHED, 139 ms]
3.47/1.79	(2) BOUNDS(1, max(2, 3 + Arg_0))
3.47/1.79	(3) Loat Proof [FINISHED, 121 ms]
3.47/1.79	(4) BOUNDS(n^1, INF)
3.47/1.79	
3.47/1.79	
3.47/1.79	----------------------------------------
3.47/1.79	
3.47/1.79	(0)
3.47/1.79	Obligation:
3.47/1.79	Complexity Int TRS consisting of the following rules:
3.47/1.79	f300(A, B) -> Com_1(f1(A, C)) :|: 0 >= A + 1
3.47/1.79	f300(A, B) -> Com_1(f300(-(1) + A, B)) :|: A >= 0
3.47/1.79	f2(A, B) -> Com_1(f300(A, B)) :|: TRUE
3.47/1.79	
3.47/1.79	The start-symbols are:[f2_2]
3.47/1.79	
3.47/1.79	
3.47/1.79	----------------------------------------
3.47/1.79	
3.47/1.79	(1) Koat2 Proof (FINISHED)
3.47/1.79	YES( ?, max([2, 3+Arg_0]) {O(n)})
3.47/1.79	
3.47/1.79	
3.47/1.79	
3.47/1.79	Initial Complexity Problem:
3.47/1.79	
3.47/1.79	  Start:  f2  
3.47/1.79	
3.47/1.79	  Program_Vars:  Arg_0, Arg_1  
3.47/1.79	
3.47/1.79	  Temp_Vars:  C  
3.47/1.79	
3.47/1.79	  Locations:  f1, f2, f300  
3.47/1.79	
3.47/1.79	  Transitions:
3.47/1.79	
3.47/1.79	  f2(Arg_0,Arg_1) -> f300(Arg_0,Arg_1):|:
3.47/1.79	
3.47/1.79	  f300(Arg_0,Arg_1) -> f1(Arg_0,C):|:Arg_0+1 <= 0
3.47/1.79	
3.47/1.79	  f300(Arg_0,Arg_1) -> f300(-1+Arg_0,Arg_1):|:0 <= Arg_0  
3.47/1.79	
3.47/1.79	
3.47/1.79	
3.47/1.79	Timebounds: 
3.47/1.79	
3.47/1.79	  Overall timebound: max([2, 3+Arg_0]) {O(n)}
3.47/1.79	
3.47/1.79	  2: f2->f300: 1 {O(1)}
3.47/1.79	
3.47/1.79	  0: f300->f1: 1 {O(1)}
3.47/1.79	
3.47/1.79	  1: f300->f300: max([0, 1+Arg_0]) {O(n)}
3.47/1.79	
3.47/1.79	
3.47/1.79	
3.47/1.79	Costbounds:
3.47/1.79	
3.47/1.79	  Overall costbound: max([2, 3+Arg_0]) {O(n)}
3.47/1.79	
3.47/1.79	  2: f2->f300: 1 {O(1)}
3.47/1.79	
3.47/1.79	  0: f300->f1: 1 {O(1)}
3.47/1.79	
3.47/1.79	  1: f300->f300: max([0, 1+Arg_0]) {O(n)}
3.47/1.79	
3.47/1.79	
3.47/1.79	
3.47/1.79	Sizebounds:
3.47/1.79	
3.47/1.79	`Lower:
3.47/1.79	
3.47/1.79	  2: f2->f300, Arg_0: Arg_0 {O(n)}
3.47/1.79	
3.47/1.79	  2: f2->f300, Arg_1: Arg_1 {O(n)}
3.47/1.79	
3.47/1.79	  0: f300->f1, Arg_0: min([-1, Arg_0]) {O(n)}
3.47/1.79	
3.47/1.79	  1: f300->f300, Arg_0: -1 {O(1)}
3.47/1.79	
3.47/1.79	  1: f300->f300, Arg_1: Arg_1 {O(n)}
3.47/1.79	
3.47/1.79	`Upper:
3.47/1.79	
3.47/1.79	  2: f2->f300, Arg_0: Arg_0 {O(n)}
3.47/1.79	
3.47/1.79	  2: f2->f300, Arg_1: Arg_1 {O(n)}
3.47/1.79	
3.47/1.79	  0: f300->f1, Arg_0: -1 {O(1)}
3.47/1.79	
3.47/1.79	  1: f300->f300, Arg_0: Arg_0 {O(n)}
3.47/1.79	
3.47/1.79	  1: f300->f300, Arg_1: Arg_1 {O(n)}
3.47/1.79	
3.47/1.79	
3.47/1.79	----------------------------------------
3.47/1.79	
3.47/1.79	(2)
3.47/1.79	BOUNDS(1, max(2, 3 + Arg_0))
3.47/1.79	
3.47/1.79	----------------------------------------
3.47/1.79	
3.47/1.79	(3) Loat Proof (FINISHED)
3.47/1.79	
3.47/1.79	
3.47/1.79	### Pre-processing the ITS problem ###
3.47/1.79	
3.47/1.79	
3.47/1.79	
3.47/1.79	Initial linear ITS problem
3.47/1.79	
3.47/1.79	   Start location: f2
3.47/1.79	
3.47/1.79	      0: f300 -> f1 : B'=free, [ 0>=1+A ], cost: 1
3.86/1.79	
3.86/1.79	      1: f300 -> f300 : A'=-1+A, [ A>=0 ], cost: 1
3.86/1.79	
3.86/1.79	      2: f2 -> f300 : [], cost: 1
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	Removed unreachable and leaf rules:
3.86/1.79	
3.86/1.79	   Start location: f2
3.86/1.79	
3.86/1.79	      1: f300 -> f300 : A'=-1+A, [ A>=0 ], cost: 1
3.86/1.79	
3.86/1.79	      2: f2 -> f300 : [], cost: 1
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	### Simplification by acceleration and chaining ###
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	Accelerating simple loops of location 0.
3.86/1.79	
3.86/1.79	   Accelerating the following rules:
3.86/1.79	
3.86/1.79	      1: f300 -> f300 : A'=-1+A, [ A>=0 ], cost: 1
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	   Accelerated rule 1 with metering function 1+A, yielding the new rule 3.
3.86/1.79	
3.86/1.79	   Removing the simple loops: 1.
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	Accelerated all simple loops using metering functions (where possible):
3.86/1.79	
3.86/1.79	   Start location: f2
3.86/1.79	
3.86/1.79	      3: f300 -> f300 : A'=-1, [ A>=0 ], cost: 1+A
3.86/1.79	
3.86/1.79	      2: f2 -> f300 : [], cost: 1
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	Chained accelerated rules (with incoming rules):
3.86/1.79	
3.86/1.79	   Start location: f2
3.86/1.79	
3.86/1.79	      2: f2 -> f300 : [], cost: 1
3.86/1.79	
3.86/1.79	      4: f2 -> f300 : A'=-1, [ A>=0 ], cost: 2+A
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	Removed unreachable locations (and leaf rules with constant cost):
3.86/1.79	
3.86/1.79	   Start location: f2
3.86/1.79	
3.86/1.79	      4: f2 -> f300 : A'=-1, [ A>=0 ], cost: 2+A
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	### Computing asymptotic complexity ###
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	Fully simplified ITS problem
3.86/1.79	
3.86/1.79	   Start location: f2
3.86/1.79	
3.86/1.79	      4: f2 -> f300 : A'=-1, [ A>=0 ], cost: 2+A
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	Computing asymptotic complexity for rule 4
3.86/1.79	
3.86/1.79	   Solved the limit problem by the following transformations:
3.86/1.79	
3.86/1.79	      Created initial limit problem:
3.86/1.79	
3.86/1.79	      2+A (+), 1+A (+/+!) [not solved]
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	      removing all constraints (solved by SMT)
3.86/1.79	
3.86/1.79	      resulting limit problem: [solved]
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	      applying transformation rule (C) using substitution {A==n}
3.86/1.79	
3.86/1.79	      resulting limit problem:
3.86/1.79	
3.86/1.79	      [solved]
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	   Solution:
3.86/1.79	
3.86/1.79	      A / n
3.86/1.79	
3.86/1.79	   Resulting cost 2+n has complexity: Poly(n^1)
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	Found new complexity Poly(n^1).
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	Obtained the following overall complexity (w.r.t. the length of the input n):
3.86/1.79	
3.86/1.79	   Complexity:  Poly(n^1)
3.86/1.79	
3.86/1.79	   Cpx degree:  1
3.86/1.79	
3.86/1.79	   Solved cost: 2+n
3.86/1.79	
3.86/1.79	   Rule cost:   2+A
3.86/1.79	
3.86/1.79	   Rule guard:  [ A>=0 ]
3.86/1.79	
3.86/1.79	
3.86/1.79	
3.86/1.79	WORST_CASE(Omega(n^1),?)
3.86/1.79	
3.86/1.79	
3.86/1.79	----------------------------------------
3.86/1.79	
3.86/1.79	(4)
3.86/1.79	BOUNDS(n^1, INF)
3.86/1.82	EOF