3.41/1.76	WORST_CASE(Omega(n^1), O(n^1))
3.41/1.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.41/1.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.41/1.77	
3.41/1.77	
3.41/1.77	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0)).
3.41/1.77	
3.41/1.77	(0) CpxIntTrs
3.41/1.77	(1) Koat2 Proof [FINISHED, 13 ms]
3.41/1.77	(2) BOUNDS(1, max(1, 1 + Arg_0))
3.41/1.77	(3) Loat Proof [FINISHED, 114 ms]
3.41/1.77	(4) BOUNDS(n^1, INF)
3.41/1.77	
3.41/1.77	
3.41/1.77	----------------------------------------
3.41/1.77	
3.41/1.77	(0)
3.41/1.77	Obligation:
3.41/1.77	Complexity Int TRS consisting of the following rules:
3.41/1.77	f3(A) -> Com_1(f1(A)) :|: A >= 0
3.41/1.77	f1(A) -> Com_1(f1(A - 1)) :|: A >= 1
3.41/1.77	
3.41/1.77	The start-symbols are:[f3_1]
3.41/1.77	
3.41/1.77	
3.41/1.77	----------------------------------------
3.41/1.77	
3.41/1.77	(1) Koat2 Proof (FINISHED)
3.41/1.77	YES( ?, max([1, 1+Arg_0]) {O(n)})
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	Initial Complexity Problem:
3.41/1.77	
3.41/1.77	  Start:  f3  
3.41/1.77	
3.41/1.77	  Program_Vars:  Arg_0  
3.41/1.77	
3.41/1.77	  Temp_Vars:    
3.41/1.77	
3.41/1.77	  Locations:  f1, f3  
3.41/1.77	
3.41/1.77	  Transitions:
3.41/1.77	
3.41/1.77	  f1(Arg_0) -> f1(Arg_0-1):|:0 <= Arg_0 && 1 <= Arg_0
3.41/1.77	
3.41/1.77	  f3(Arg_0) -> f1(Arg_0):|:0 <= Arg_0  
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	Timebounds: 
3.41/1.77	
3.41/1.77	  Overall timebound: max([1, 1+Arg_0]) {O(n)}
3.41/1.77	
3.41/1.77	  1: f1->f1: max([0, Arg_0]) {O(n)}
3.41/1.77	
3.41/1.77	  0: f3->f1: 1 {O(1)}
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	Costbounds:
3.41/1.77	
3.41/1.77	  Overall costbound: max([1, 1+Arg_0]) {O(n)}
3.41/1.77	
3.41/1.77	  1: f1->f1: max([0, Arg_0]) {O(n)}
3.41/1.77	
3.41/1.77	  0: f3->f1: 1 {O(1)}
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	Sizebounds:
3.41/1.77	
3.41/1.77	`Lower:
3.41/1.77	
3.41/1.77	  1: f1->f1, Arg_0: 0 {O(1)}
3.41/1.77	
3.41/1.77	  0: f3->f1, Arg_0: 0 {O(1)}
3.41/1.77	
3.41/1.77	`Upper:
3.41/1.77	
3.41/1.77	  1: f1->f1, Arg_0: Arg_0 {O(n)}
3.41/1.77	
3.41/1.77	  0: f3->f1, Arg_0: Arg_0 {O(n)}
3.41/1.77	
3.41/1.77	
3.41/1.77	----------------------------------------
3.41/1.77	
3.41/1.77	(2)
3.41/1.77	BOUNDS(1, max(1, 1 + Arg_0))
3.41/1.77	
3.41/1.77	----------------------------------------
3.41/1.77	
3.41/1.77	(3) Loat Proof (FINISHED)
3.41/1.77	
3.41/1.77	
3.41/1.77	### Pre-processing the ITS problem ###
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	Initial linear ITS problem
3.41/1.77	
3.41/1.77	   Start location: f3
3.41/1.77	
3.41/1.77	      0: f3 -> f1 : [ A>=0 ], cost: 1
3.41/1.77	
3.41/1.77	      1: f1 -> f1 : A'=-1+A, [ A>=1 ], cost: 1
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	### Simplification by acceleration and chaining ###
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	Accelerating simple loops of location 1.
3.41/1.77	
3.41/1.77	   Accelerating the following rules:
3.41/1.77	
3.41/1.77	      1: f1 -> f1 : A'=-1+A, [ A>=1 ], cost: 1
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	   Accelerated rule 1 with metering function A, yielding the new rule 2.
3.41/1.77	
3.41/1.77	   Removing the simple loops: 1.
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	Accelerated all simple loops using metering functions (where possible):
3.41/1.77	
3.41/1.77	   Start location: f3
3.41/1.77	
3.41/1.77	      0: f3 -> f1 : [ A>=0 ], cost: 1
3.41/1.77	
3.41/1.77	      2: f1 -> f1 : A'=0, [ A>=1 ], cost: A
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	Chained accelerated rules (with incoming rules):
3.41/1.77	
3.41/1.77	   Start location: f3
3.41/1.77	
3.41/1.77	      0: f3 -> f1 : [ A>=0 ], cost: 1
3.41/1.77	
3.41/1.77	      3: f3 -> f1 : A'=0, [ A>=1 ], cost: 1+A
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	Removed unreachable locations (and leaf rules with constant cost):
3.41/1.77	
3.41/1.77	   Start location: f3
3.41/1.77	
3.41/1.77	      3: f3 -> f1 : A'=0, [ A>=1 ], cost: 1+A
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	### Computing asymptotic complexity ###
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	Fully simplified ITS problem
3.41/1.77	
3.41/1.77	   Start location: f3
3.41/1.77	
3.41/1.77	      3: f3 -> f1 : A'=0, [ A>=1 ], cost: 1+A
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	Computing asymptotic complexity for rule 3
3.41/1.77	
3.41/1.77	   Solved the limit problem by the following transformations:
3.41/1.77	
3.41/1.77	      Created initial limit problem:
3.41/1.77	
3.41/1.77	      A (+/+!), 1+A (+) [not solved]
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	      removing all constraints (solved by SMT)
3.41/1.77	
3.41/1.77	      resulting limit problem: [solved]
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	      applying transformation rule (C) using substitution {A==n}
3.41/1.77	
3.41/1.77	      resulting limit problem:
3.41/1.77	
3.41/1.77	      [solved]
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	   Solution:
3.41/1.77	
3.41/1.77	      A / n
3.41/1.77	
3.41/1.77	   Resulting cost 1+n has complexity: Poly(n^1)
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	Found new complexity Poly(n^1).
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	Obtained the following overall complexity (w.r.t. the length of the input n):
3.41/1.77	
3.41/1.77	   Complexity:  Poly(n^1)
3.41/1.77	
3.41/1.77	   Cpx degree:  1
3.41/1.77	
3.41/1.77	   Solved cost: 1+n
3.41/1.77	
3.41/1.77	   Rule cost:   1+A
3.41/1.77	
3.41/1.77	   Rule guard:  [ A>=1 ]
3.41/1.77	
3.41/1.77	
3.41/1.77	
3.41/1.77	WORST_CASE(Omega(n^1),?)
3.41/1.77	
3.41/1.77	
3.41/1.77	----------------------------------------
3.41/1.77	
3.41/1.77	(4)
3.41/1.77	BOUNDS(n^1, INF)
3.62/1.79	EOF