3.55/1.75	WORST_CASE(Omega(n^1), O(n^1))
3.55/1.76	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
3.55/1.76	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.55/1.76	
3.55/1.76	
3.55/1.76	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(2, 3 + Arg_0)).
3.55/1.76	
3.55/1.76	(0) CpxIntTrs
3.55/1.76	(1) Koat2 Proof [FINISHED, 143 ms]
3.55/1.76	(2) BOUNDS(1, max(2, 3 + Arg_0))
3.55/1.76	(3) Loat Proof [FINISHED, 115 ms]
3.55/1.76	(4) BOUNDS(n^1, INF)
3.55/1.76	
3.55/1.76	
3.55/1.76	----------------------------------------
3.55/1.76	
3.55/1.76	(0)
3.55/1.76	Obligation:
3.55/1.76	Complexity Int TRS consisting of the following rules:
3.55/1.76	f3(A, B) -> Com_1(f1(A, B)) :|: TRUE
3.55/1.76	f1(A, B) -> Com_1(f1(-(1) + A, B)) :|: A >= 0
3.55/1.76	f1(A, B) -> Com_1(f300(A, C)) :|: 0 >= A + 1
3.55/1.76	
3.55/1.76	The start-symbols are:[f3_2]
3.55/1.76	
3.55/1.76	
3.55/1.76	----------------------------------------
3.55/1.76	
3.55/1.76	(1) Koat2 Proof (FINISHED)
3.55/1.76	YES( ?, max([2, 3+Arg_0]) {O(n)})
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Initial Complexity Problem:
3.55/1.76	
3.55/1.76	  Start:  f3  
3.55/1.76	
3.55/1.76	  Program_Vars:  Arg_0, Arg_1  
3.55/1.76	
3.55/1.76	  Temp_Vars:  C  
3.55/1.76	
3.55/1.76	  Locations:  f1, f3, f300  
3.55/1.76	
3.55/1.76	  Transitions:
3.55/1.76	
3.55/1.76	  f1(Arg_0,Arg_1) -> f1(-1+Arg_0,Arg_1):|:0 <= Arg_0
3.55/1.76	
3.55/1.76	  f1(Arg_0,Arg_1) -> f300(Arg_0,C):|:Arg_0+1 <= 0
3.55/1.76	
3.55/1.76	  f3(Arg_0,Arg_1) -> f1(Arg_0,Arg_1):|:  
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Timebounds: 
3.55/1.76	
3.55/1.76	  Overall timebound: max([2, 3+Arg_0]) {O(n)}
3.55/1.76	
3.55/1.76	  1: f1->f1: max([0, 1+Arg_0]) {O(n)}
3.55/1.76	
3.55/1.76	  2: f1->f300: 1 {O(1)}
3.55/1.76	
3.55/1.76	  0: f3->f1: 1 {O(1)}
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Costbounds:
3.55/1.76	
3.55/1.76	  Overall costbound: max([2, 3+Arg_0]) {O(n)}
3.55/1.76	
3.55/1.76	  1: f1->f1: max([0, 1+Arg_0]) {O(n)}
3.55/1.76	
3.55/1.76	  2: f1->f300: 1 {O(1)}
3.55/1.76	
3.55/1.76	  0: f3->f1: 1 {O(1)}
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Sizebounds:
3.55/1.76	
3.55/1.76	`Lower:
3.55/1.76	
3.55/1.76	  1: f1->f1, Arg_0: -1 {O(1)}
3.55/1.76	
3.55/1.76	  1: f1->f1, Arg_1: Arg_1 {O(n)}
3.55/1.76	
3.55/1.76	  2: f1->f300, Arg_0: min([-1, Arg_0]) {O(n)}
3.55/1.76	
3.55/1.76	  0: f3->f1, Arg_0: Arg_0 {O(n)}
3.55/1.76	
3.55/1.76	  0: f3->f1, Arg_1: Arg_1 {O(n)}
3.55/1.76	
3.55/1.76	`Upper:
3.55/1.76	
3.55/1.76	  1: f1->f1, Arg_0: Arg_0 {O(n)}
3.55/1.76	
3.55/1.76	  1: f1->f1, Arg_1: Arg_1 {O(n)}
3.55/1.76	
3.55/1.76	  2: f1->f300, Arg_0: -1 {O(1)}
3.55/1.76	
3.55/1.76	  0: f3->f1, Arg_0: Arg_0 {O(n)}
3.55/1.76	
3.55/1.76	  0: f3->f1, Arg_1: Arg_1 {O(n)}
3.55/1.76	
3.55/1.76	
3.55/1.76	----------------------------------------
3.55/1.76	
3.55/1.76	(2)
3.55/1.76	BOUNDS(1, max(2, 3 + Arg_0))
3.55/1.76	
3.55/1.76	----------------------------------------
3.55/1.76	
3.55/1.76	(3) Loat Proof (FINISHED)
3.55/1.76	
3.55/1.76	
3.55/1.76	### Pre-processing the ITS problem ###
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Initial linear ITS problem
3.55/1.76	
3.55/1.76	   Start location: f3
3.55/1.76	
3.55/1.76	      0: f3 -> f1 : [], cost: 1
3.55/1.76	
3.55/1.76	      1: f1 -> f1 : A'=-1+A, [ A>=0 ], cost: 1
3.55/1.76	
3.55/1.76	      2: f1 -> f300 : B'=free, [ 0>=1+A ], cost: 1
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Removed unreachable and leaf rules:
3.55/1.76	
3.55/1.76	   Start location: f3
3.55/1.76	
3.55/1.76	      0: f3 -> f1 : [], cost: 1
3.55/1.76	
3.55/1.76	      1: f1 -> f1 : A'=-1+A, [ A>=0 ], cost: 1
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	### Simplification by acceleration and chaining ###
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Accelerating simple loops of location 1.
3.55/1.76	
3.55/1.76	   Accelerating the following rules:
3.55/1.76	
3.55/1.76	      1: f1 -> f1 : A'=-1+A, [ A>=0 ], cost: 1
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	   Accelerated rule 1 with metering function 1+A, yielding the new rule 3.
3.55/1.76	
3.55/1.76	   Removing the simple loops: 1.
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Accelerated all simple loops using metering functions (where possible):
3.55/1.76	
3.55/1.76	   Start location: f3
3.55/1.76	
3.55/1.76	      0: f3 -> f1 : [], cost: 1
3.55/1.76	
3.55/1.76	      3: f1 -> f1 : A'=-1, [ A>=0 ], cost: 1+A
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Chained accelerated rules (with incoming rules):
3.55/1.76	
3.55/1.76	   Start location: f3
3.55/1.76	
3.55/1.76	      0: f3 -> f1 : [], cost: 1
3.55/1.76	
3.55/1.76	      4: f3 -> f1 : A'=-1, [ A>=0 ], cost: 2+A
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Removed unreachable locations (and leaf rules with constant cost):
3.55/1.76	
3.55/1.76	   Start location: f3
3.55/1.76	
3.55/1.76	      4: f3 -> f1 : A'=-1, [ A>=0 ], cost: 2+A
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	### Computing asymptotic complexity ###
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Fully simplified ITS problem
3.55/1.76	
3.55/1.76	   Start location: f3
3.55/1.76	
3.55/1.76	      4: f3 -> f1 : A'=-1, [ A>=0 ], cost: 2+A
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Computing asymptotic complexity for rule 4
3.55/1.76	
3.55/1.76	   Solved the limit problem by the following transformations:
3.55/1.76	
3.55/1.76	      Created initial limit problem:
3.55/1.76	
3.55/1.76	      2+A (+), 1+A (+/+!) [not solved]
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	      removing all constraints (solved by SMT)
3.55/1.76	
3.55/1.76	      resulting limit problem: [solved]
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	      applying transformation rule (C) using substitution {A==n}
3.55/1.76	
3.55/1.76	      resulting limit problem:
3.55/1.76	
3.55/1.76	      [solved]
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	   Solution:
3.55/1.76	
3.55/1.76	      A / n
3.55/1.76	
3.55/1.76	   Resulting cost 2+n has complexity: Poly(n^1)
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Found new complexity Poly(n^1).
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	Obtained the following overall complexity (w.r.t. the length of the input n):
3.55/1.76	
3.55/1.76	   Complexity:  Poly(n^1)
3.55/1.76	
3.55/1.76	   Cpx degree:  1
3.55/1.76	
3.55/1.76	   Solved cost: 2+n
3.55/1.76	
3.55/1.76	   Rule cost:   2+A
3.55/1.76	
3.55/1.76	   Rule guard:  [ A>=0 ]
3.55/1.76	
3.55/1.76	
3.55/1.76	
3.55/1.76	WORST_CASE(Omega(n^1),?)
3.55/1.76	
3.55/1.76	
3.55/1.76	----------------------------------------
3.55/1.76	
3.55/1.76	(4)
3.55/1.76	BOUNDS(n^1, INF)
3.55/1.78	EOF