3.61/1.85	WORST_CASE(Omega(n^1), O(n^1))
3.61/1.86	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.61/1.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.61/1.86	
3.61/1.86	
3.61/1.86	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(2, 2 + Arg_0)).
3.61/1.86	
3.61/1.86	(0) CpxIntTrs
3.61/1.86	(1) Koat2 Proof [FINISHED, 29 ms]
3.61/1.86	(2) BOUNDS(1, max(2, 2 + Arg_0))
3.61/1.86	(3) Loat Proof [FINISHED, 104 ms]
3.61/1.86	(4) BOUNDS(n^1, INF)
3.61/1.86	
3.61/1.86	
3.61/1.86	----------------------------------------
3.61/1.86	
3.61/1.86	(0)
3.61/1.86	Obligation:
3.61/1.86	Complexity Int TRS consisting of the following rules:
3.61/1.86	f2(A, B) -> Com_1(f3(A, 1)) :|: 0 >= A
3.61/1.86	f0(A, B) -> Com_1(f2(A, 0)) :|: TRUE
3.61/1.86	f2(A, B) -> Com_1(f2(A - 1, B)) :|: A >= 1
3.61/1.86	
3.61/1.86	The start-symbols are:[f0_2]
3.61/1.86	
3.61/1.86	
3.61/1.86	----------------------------------------
3.61/1.86	
3.61/1.86	(1) Koat2 Proof (FINISHED)
3.61/1.86	YES( ?, max([2, 2+Arg_0]) {O(n)})
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Initial Complexity Problem:
3.61/1.86	
3.61/1.86	  Start:  f0  
3.61/1.86	
3.61/1.86	  Program_Vars:  Arg_0, Arg_1  
3.61/1.86	
3.61/1.86	  Temp_Vars:    
3.61/1.86	
3.61/1.86	  Locations:  f0, f2, f3  
3.61/1.86	
3.61/1.86	  Transitions:
3.61/1.86	
3.61/1.86	  f0(Arg_0,Arg_1) -> f2(Arg_0,0):|:
3.61/1.86	
3.61/1.86	  f2(Arg_0,Arg_1) -> f2(Arg_0-1,Arg_1):|:Arg_1 <= 0 && 0 <= Arg_1 && 1 <= Arg_0
3.61/1.86	
3.61/1.86	  f2(Arg_0,Arg_1) -> f3(Arg_0,1):|:Arg_1 <= 0 && 0 <= Arg_1 && Arg_0 <= 0  
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Timebounds: 
3.61/1.86	
3.61/1.86	  Overall timebound: max([2, 2+Arg_0]) {O(n)}
3.61/1.86	
3.61/1.86	  1: f0->f2: 1 {O(1)}
3.61/1.86	
3.61/1.86	  0: f2->f3: 1 {O(1)}
3.61/1.86	
3.61/1.86	  2: f2->f2: max([0, Arg_0]) {O(n)}
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Costbounds:
3.61/1.86	
3.61/1.86	  Overall costbound: max([2, 2+Arg_0]) {O(n)}
3.61/1.86	
3.61/1.86	  1: f0->f2: 1 {O(1)}
3.61/1.86	
3.61/1.86	  0: f2->f3: 1 {O(1)}
3.61/1.86	
3.61/1.86	  2: f2->f2: max([0, Arg_0]) {O(n)}
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Sizebounds:
3.61/1.86	
3.61/1.86	`Lower:
3.61/1.86	
3.61/1.86	  1: f0->f2, Arg_0: Arg_0 {O(n)}
3.61/1.86	
3.61/1.86	  1: f0->f2, Arg_1: 0 {O(1)}
3.61/1.86	
3.61/1.86	  0: f2->f3, Arg_0: min([0, Arg_0]) {O(n)}
3.61/1.86	
3.61/1.86	  0: f2->f3, Arg_1: 1 {O(1)}
3.61/1.86	
3.61/1.86	  2: f2->f2, Arg_0: 0 {O(1)}
3.61/1.86	
3.61/1.86	  2: f2->f2, Arg_1: 0 {O(1)}
3.61/1.86	
3.61/1.86	`Upper:
3.61/1.86	
3.61/1.86	  1: f0->f2, Arg_0: Arg_0 {O(n)}
3.61/1.86	
3.61/1.86	  1: f0->f2, Arg_1: 0 {O(1)}
3.61/1.86	
3.61/1.86	  0: f2->f3, Arg_0: 0 {O(1)}
3.61/1.86	
3.61/1.86	  0: f2->f3, Arg_1: 1 {O(1)}
3.61/1.86	
3.61/1.86	  2: f2->f2, Arg_0: Arg_0 {O(n)}
3.61/1.86	
3.61/1.86	  2: f2->f2, Arg_1: 0 {O(1)}
3.61/1.86	
3.61/1.86	
3.61/1.86	----------------------------------------
3.61/1.86	
3.61/1.86	(2)
3.61/1.86	BOUNDS(1, max(2, 2 + Arg_0))
3.61/1.86	
3.61/1.86	----------------------------------------
3.61/1.86	
3.61/1.86	(3) Loat Proof (FINISHED)
3.61/1.86	
3.61/1.86	
3.61/1.86	### Pre-processing the ITS problem ###
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Initial linear ITS problem
3.61/1.86	
3.61/1.86	   Start location: f0
3.61/1.86	
3.61/1.86	      0: f2 -> f3 : B'=1, [ 0>=A ], cost: 1
3.61/1.86	
3.61/1.86	      2: f2 -> f2 : A'=-1+A, [ A>=1 ], cost: 1
3.61/1.86	
3.61/1.86	      1: f0 -> f2 : B'=0, [], cost: 1
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Removed unreachable and leaf rules:
3.61/1.86	
3.61/1.86	   Start location: f0
3.61/1.86	
3.61/1.86	      2: f2 -> f2 : A'=-1+A, [ A>=1 ], cost: 1
3.61/1.86	
3.61/1.86	      1: f0 -> f2 : B'=0, [], cost: 1
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	### Simplification by acceleration and chaining ###
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Accelerating simple loops of location 0.
3.61/1.86	
3.61/1.86	   Accelerating the following rules:
3.61/1.86	
3.61/1.86	      2: f2 -> f2 : A'=-1+A, [ A>=1 ], cost: 1
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	   Accelerated rule 2 with metering function A, yielding the new rule 3.
3.61/1.86	
3.61/1.86	   Removing the simple loops: 2.
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Accelerated all simple loops using metering functions (where possible):
3.61/1.86	
3.61/1.86	   Start location: f0
3.61/1.86	
3.61/1.86	      3: f2 -> f2 : A'=0, [ A>=1 ], cost: A
3.61/1.86	
3.61/1.86	      1: f0 -> f2 : B'=0, [], cost: 1
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Chained accelerated rules (with incoming rules):
3.61/1.86	
3.61/1.86	   Start location: f0
3.61/1.86	
3.61/1.86	      1: f0 -> f2 : B'=0, [], cost: 1
3.61/1.86	
3.61/1.86	      4: f0 -> f2 : A'=0, B'=0, [ A>=1 ], cost: 1+A
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Removed unreachable locations (and leaf rules with constant cost):
3.61/1.86	
3.61/1.86	   Start location: f0
3.61/1.86	
3.61/1.86	      4: f0 -> f2 : A'=0, B'=0, [ A>=1 ], cost: 1+A
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	### Computing asymptotic complexity ###
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Fully simplified ITS problem
3.61/1.86	
3.61/1.86	   Start location: f0
3.61/1.86	
3.61/1.86	      4: f0 -> f2 : A'=0, B'=0, [ A>=1 ], cost: 1+A
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Computing asymptotic complexity for rule 4
3.61/1.86	
3.61/1.86	   Solved the limit problem by the following transformations:
3.61/1.86	
3.61/1.86	      Created initial limit problem:
3.61/1.86	
3.61/1.86	      A (+/+!), 1+A (+) [not solved]
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	      removing all constraints (solved by SMT)
3.61/1.86	
3.61/1.86	      resulting limit problem: [solved]
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	      applying transformation rule (C) using substitution {A==n}
3.61/1.86	
3.61/1.86	      resulting limit problem:
3.61/1.86	
3.61/1.86	      [solved]
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	   Solution:
3.61/1.86	
3.61/1.86	      A / n
3.61/1.86	
3.61/1.86	   Resulting cost 1+n has complexity: Poly(n^1)
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Found new complexity Poly(n^1).
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	Obtained the following overall complexity (w.r.t. the length of the input n):
3.61/1.86	
3.61/1.86	   Complexity:  Poly(n^1)
3.61/1.86	
3.61/1.86	   Cpx degree:  1
3.61/1.86	
3.61/1.86	   Solved cost: 1+n
3.61/1.86	
3.61/1.86	   Rule cost:   1+A
3.61/1.86	
3.61/1.86	   Rule guard:  [ A>=1 ]
3.61/1.86	
3.61/1.86	
3.61/1.86	
3.61/1.86	WORST_CASE(Omega(n^1),?)
3.61/1.86	
3.61/1.86	
3.61/1.86	----------------------------------------
3.61/1.86	
3.61/1.86	(4)
3.61/1.86	BOUNDS(n^1, INF)
3.61/1.89	EOF