3.74/1.84	WORST_CASE(Omega(n^1), O(n^1))
3.74/1.85	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.74/1.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.74/1.85	
3.74/1.85	
3.74/1.85	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(2, 2 + Arg_0)).
3.74/1.85	
3.74/1.85	(0) CpxIntTrs
3.74/1.85	(1) Koat2 Proof [FINISHED, 136 ms]
3.74/1.85	(2) BOUNDS(1, max(2, 2 + Arg_0))
3.74/1.85	(3) Loat Proof [FINISHED, 131 ms]
3.74/1.85	(4) BOUNDS(n^1, INF)
3.74/1.85	
3.74/1.85	
3.74/1.85	----------------------------------------
3.74/1.85	
3.74/1.85	(0)
3.74/1.85	Obligation:
3.74/1.85	Complexity Int TRS consisting of the following rules:
3.74/1.85	f2(A, B) -> Com_1(f1(-(1) + A, C)) :|: 0 >= A
3.74/1.85	f2(A, B) -> Com_1(f2(-(1) + A, B)) :|: A >= 1
3.74/1.85	f300(A, B) -> Com_1(f2(A, B)) :|: TRUE
3.74/1.85	
3.74/1.85	The start-symbols are:[f300_2]
3.74/1.85	
3.74/1.85	
3.74/1.85	----------------------------------------
3.74/1.85	
3.74/1.85	(1) Koat2 Proof (FINISHED)
3.74/1.85	YES( ?, max([2, 2+Arg_0]) {O(n)})
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Initial Complexity Problem:
3.74/1.85	
3.74/1.85	  Start:  f300  
3.74/1.85	
3.74/1.85	  Program_Vars:  Arg_0, Arg_1  
3.74/1.85	
3.74/1.85	  Temp_Vars:  C  
3.74/1.85	
3.74/1.85	  Locations:  f1, f2, f300  
3.74/1.85	
3.74/1.85	  Transitions:
3.74/1.85	
3.74/1.85	  f2(Arg_0,Arg_1) -> f1(-1+Arg_0,C):|:Arg_0 <= 0
3.74/1.85	
3.74/1.85	  f2(Arg_0,Arg_1) -> f2(-1+Arg_0,Arg_1):|:1 <= Arg_0
3.74/1.85	
3.74/1.85	  f300(Arg_0,Arg_1) -> f2(Arg_0,Arg_1):|:  
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Timebounds: 
3.74/1.85	
3.74/1.85	  Overall timebound: max([2, 2+Arg_0]) {O(n)}
3.74/1.85	
3.74/1.85	  0: f2->f1: 1 {O(1)}
3.74/1.85	
3.74/1.85	  1: f2->f2: max([0, Arg_0]) {O(n)}
3.74/1.85	
3.74/1.85	  2: f300->f2: 1 {O(1)}
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Costbounds:
3.74/1.85	
3.74/1.85	  Overall costbound: max([2, 2+Arg_0]) {O(n)}
3.74/1.85	
3.74/1.85	  0: f2->f1: 1 {O(1)}
3.74/1.85	
3.74/1.85	  1: f2->f2: max([0, Arg_0]) {O(n)}
3.74/1.85	
3.74/1.85	  2: f300->f2: 1 {O(1)}
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Sizebounds:
3.74/1.85	
3.74/1.85	`Lower:
3.74/1.85	
3.74/1.85	  0: f2->f1, Arg_0: min([-1, -(1-Arg_0)]) {O(n)}
3.74/1.85	
3.74/1.85	  1: f2->f2, Arg_0: 0 {O(1)}
3.74/1.85	
3.74/1.85	  1: f2->f2, Arg_1: Arg_1 {O(n)}
3.74/1.85	
3.74/1.85	  2: f300->f2, Arg_0: Arg_0 {O(n)}
3.74/1.85	
3.74/1.85	  2: f300->f2, Arg_1: Arg_1 {O(n)}
3.74/1.85	
3.74/1.85	`Upper:
3.74/1.85	
3.74/1.85	  0: f2->f1, Arg_0: -1 {O(1)}
3.74/1.85	
3.74/1.85	  1: f2->f2, Arg_0: Arg_0 {O(n)}
3.74/1.85	
3.74/1.85	  1: f2->f2, Arg_1: Arg_1 {O(n)}
3.74/1.85	
3.74/1.85	  2: f300->f2, Arg_0: Arg_0 {O(n)}
3.74/1.85	
3.74/1.85	  2: f300->f2, Arg_1: Arg_1 {O(n)}
3.74/1.85	
3.74/1.85	
3.74/1.85	----------------------------------------
3.74/1.85	
3.74/1.85	(2)
3.74/1.85	BOUNDS(1, max(2, 2 + Arg_0))
3.74/1.85	
3.74/1.85	----------------------------------------
3.74/1.85	
3.74/1.85	(3) Loat Proof (FINISHED)
3.74/1.85	
3.74/1.85	
3.74/1.85	### Pre-processing the ITS problem ###
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Initial linear ITS problem
3.74/1.85	
3.74/1.85	   Start location: f300
3.74/1.85	
3.74/1.85	      0: f2 -> f1 : A'=-1+A, B'=free, [ 0>=A ], cost: 1
3.74/1.85	
3.74/1.85	      1: f2 -> f2 : A'=-1+A, [ A>=1 ], cost: 1
3.74/1.85	
3.74/1.85	      2: f300 -> f2 : [], cost: 1
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Removed unreachable and leaf rules:
3.74/1.85	
3.74/1.85	   Start location: f300
3.74/1.85	
3.74/1.85	      1: f2 -> f2 : A'=-1+A, [ A>=1 ], cost: 1
3.74/1.85	
3.74/1.85	      2: f300 -> f2 : [], cost: 1
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	### Simplification by acceleration and chaining ###
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Accelerating simple loops of location 0.
3.74/1.85	
3.74/1.85	   Accelerating the following rules:
3.74/1.85	
3.74/1.85	      1: f2 -> f2 : A'=-1+A, [ A>=1 ], cost: 1
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	   Accelerated rule 1 with metering function A, yielding the new rule 3.
3.74/1.85	
3.74/1.85	   Removing the simple loops: 1.
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Accelerated all simple loops using metering functions (where possible):
3.74/1.85	
3.74/1.85	   Start location: f300
3.74/1.85	
3.74/1.85	      3: f2 -> f2 : A'=0, [ A>=1 ], cost: A
3.74/1.85	
3.74/1.85	      2: f300 -> f2 : [], cost: 1
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Chained accelerated rules (with incoming rules):
3.74/1.85	
3.74/1.85	   Start location: f300
3.74/1.85	
3.74/1.85	      2: f300 -> f2 : [], cost: 1
3.74/1.85	
3.74/1.85	      4: f300 -> f2 : A'=0, [ A>=1 ], cost: 1+A
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Removed unreachable locations (and leaf rules with constant cost):
3.74/1.85	
3.74/1.85	   Start location: f300
3.74/1.85	
3.74/1.85	      4: f300 -> f2 : A'=0, [ A>=1 ], cost: 1+A
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	### Computing asymptotic complexity ###
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Fully simplified ITS problem
3.74/1.85	
3.74/1.85	   Start location: f300
3.74/1.85	
3.74/1.85	      4: f300 -> f2 : A'=0, [ A>=1 ], cost: 1+A
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Computing asymptotic complexity for rule 4
3.74/1.85	
3.74/1.85	   Solved the limit problem by the following transformations:
3.74/1.85	
3.74/1.85	      Created initial limit problem:
3.74/1.85	
3.74/1.85	      A (+/+!), 1+A (+) [not solved]
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	      removing all constraints (solved by SMT)
3.74/1.85	
3.74/1.85	      resulting limit problem: [solved]
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	      applying transformation rule (C) using substitution {A==n}
3.74/1.85	
3.74/1.85	      resulting limit problem:
3.74/1.85	
3.74/1.85	      [solved]
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	   Solution:
3.74/1.85	
3.74/1.85	      A / n
3.74/1.85	
3.74/1.85	   Resulting cost 1+n has complexity: Poly(n^1)
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Found new complexity Poly(n^1).
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	Obtained the following overall complexity (w.r.t. the length of the input n):
3.74/1.85	
3.74/1.85	   Complexity:  Poly(n^1)
3.74/1.85	
3.74/1.85	   Cpx degree:  1
3.74/1.85	
3.74/1.85	   Solved cost: 1+n
3.74/1.85	
3.74/1.85	   Rule cost:   1+A
3.74/1.85	
3.74/1.85	   Rule guard:  [ A>=1 ]
3.74/1.85	
3.74/1.85	
3.74/1.85	
3.74/1.85	WORST_CASE(Omega(n^1),?)
3.74/1.85	
3.74/1.85	
3.74/1.85	----------------------------------------
3.74/1.85	
3.74/1.85	(4)
3.74/1.85	BOUNDS(n^1, INF)
3.74/1.87	EOF