3.68/1.82	WORST_CASE(Omega(n^1), O(n^1))
3.68/1.83	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.68/1.83	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.68/1.83	
3.68/1.83	
3.68/1.83	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(3, 2 + Arg_0)).
3.68/1.83	
3.68/1.83	(0) CpxIntTrs
3.68/1.83	(1) Koat2 Proof [FINISHED, 146 ms]
3.68/1.83	(2) BOUNDS(1, max(3, 2 + Arg_0))
3.68/1.83	(3) Loat Proof [FINISHED, 117 ms]
3.68/1.83	(4) BOUNDS(n^1, INF)
3.68/1.83	
3.68/1.83	
3.68/1.83	----------------------------------------
3.68/1.83	
3.68/1.83	(0)
3.68/1.83	Obligation:
3.68/1.83	Complexity Int TRS consisting of the following rules:
3.68/1.83	f5(A, B) -> Com_1(f5(-(1) + A, B)) :|: A >= 1
3.68/1.83	f5(A, B) -> Com_1(f1(A, C)) :|: 0 >= A
3.68/1.83	f300(A, B) -> Com_1(f5(-(1) + A, B)) :|: A >= 1
3.68/1.83	f300(A, B) -> Com_1(f1(A, C)) :|: 0 >= A
3.68/1.83	
3.68/1.83	The start-symbols are:[f300_2]
3.68/1.83	
3.68/1.83	
3.68/1.83	----------------------------------------
3.68/1.83	
3.68/1.83	(1) Koat2 Proof (FINISHED)
3.68/1.83	YES( ?, max([3, 2+Arg_0]) {O(n)})
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Initial Complexity Problem:
3.68/1.83	
3.68/1.83	  Start:  f300  
3.68/1.83	
3.68/1.83	  Program_Vars:  Arg_0, Arg_1  
3.68/1.83	
3.68/1.83	  Temp_Vars:  C  
3.68/1.83	
3.68/1.83	  Locations:  f1, f300, f5  
3.68/1.83	
3.68/1.83	  Transitions:
3.68/1.83	
3.68/1.83	  f300(Arg_0,Arg_1) -> f1(Arg_0,C):|:Arg_0 <= 0
3.68/1.83	
3.68/1.83	  f300(Arg_0,Arg_1) -> f5(-1+Arg_0,Arg_1):|:1 <= Arg_0
3.68/1.83	
3.68/1.83	  f5(Arg_0,Arg_1) -> f1(Arg_0,C):|:0 <= Arg_0 && Arg_0 <= 0
3.68/1.83	
3.68/1.83	  f5(Arg_0,Arg_1) -> f5(-1+Arg_0,Arg_1):|:0 <= Arg_0 && 1 <= Arg_0  
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Timebounds: 
3.68/1.83	
3.68/1.83	  Overall timebound: max([3, 2+Arg_0]) {O(n)}
3.68/1.83	
3.68/1.83	  2: f300->f5: 1 {O(1)}
3.68/1.83	
3.68/1.83	  3: f300->f1: 1 {O(1)}
3.68/1.83	
3.68/1.83	  0: f5->f5: max([0, -1+Arg_0]) {O(n)}
3.68/1.83	
3.68/1.83	  1: f5->f1: 1 {O(1)}
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Costbounds:
3.68/1.83	
3.68/1.83	  Overall costbound: max([3, 2+Arg_0]) {O(n)}
3.68/1.83	
3.68/1.83	  2: f300->f5: 1 {O(1)}
3.68/1.83	
3.68/1.83	  3: f300->f1: 1 {O(1)}
3.68/1.83	
3.68/1.83	  0: f5->f5: max([0, -1+Arg_0]) {O(n)}
3.68/1.83	
3.68/1.83	  1: f5->f1: 1 {O(1)}
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Sizebounds:
3.68/1.83	
3.68/1.83	`Lower:
3.68/1.83	
3.68/1.83	  2: f300->f5, Arg_0: 0 {O(1)}
3.68/1.83	
3.68/1.83	  2: f300->f5, Arg_1: Arg_1 {O(n)}
3.68/1.83	
3.68/1.83	  3: f300->f1, Arg_0: Arg_0 {O(n)}
3.68/1.83	
3.68/1.83	  0: f5->f5, Arg_0: 0 {O(1)}
3.68/1.83	
3.68/1.83	  0: f5->f5, Arg_1: Arg_1 {O(n)}
3.68/1.83	
3.68/1.83	  1: f5->f1, Arg_0: 0 {O(1)}
3.68/1.83	
3.68/1.83	`Upper:
3.68/1.83	
3.68/1.83	  2: f300->f5, Arg_0: -1+Arg_0 {O(n)}
3.68/1.83	
3.68/1.83	  2: f300->f5, Arg_1: Arg_1 {O(n)}
3.68/1.83	
3.68/1.83	  3: f300->f1, Arg_0: 0 {O(1)}
3.68/1.83	
3.68/1.83	  0: f5->f5, Arg_0: -1+Arg_0 {O(n)}
3.68/1.83	
3.68/1.83	  0: f5->f5, Arg_1: Arg_1 {O(n)}
3.68/1.83	
3.68/1.83	  1: f5->f1, Arg_0: 0 {O(1)}
3.68/1.83	
3.68/1.83	
3.68/1.83	----------------------------------------
3.68/1.83	
3.68/1.83	(2)
3.68/1.83	BOUNDS(1, max(3, 2 + Arg_0))
3.68/1.83	
3.68/1.83	----------------------------------------
3.68/1.83	
3.68/1.83	(3) Loat Proof (FINISHED)
3.68/1.83	
3.68/1.83	
3.68/1.83	### Pre-processing the ITS problem ###
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Initial linear ITS problem
3.68/1.83	
3.68/1.83	   Start location: f300
3.68/1.83	
3.68/1.83	      0: f5 -> f5 : A'=-1+A, [ A>=1 ], cost: 1
3.68/1.83	
3.68/1.83	      1: f5 -> f1 : B'=free, [ 0>=A ], cost: 1
3.68/1.83	
3.68/1.83	      2: f300 -> f5 : A'=-1+A, [ A>=1 ], cost: 1
3.68/1.83	
3.68/1.83	      3: f300 -> f1 : B'=free_1, [ 0>=A ], cost: 1
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Removed unreachable and leaf rules:
3.68/1.83	
3.68/1.83	   Start location: f300
3.68/1.83	
3.68/1.83	      0: f5 -> f5 : A'=-1+A, [ A>=1 ], cost: 1
3.68/1.83	
3.68/1.83	      2: f300 -> f5 : A'=-1+A, [ A>=1 ], cost: 1
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	### Simplification by acceleration and chaining ###
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Accelerating simple loops of location 0.
3.68/1.83	
3.68/1.83	   Accelerating the following rules:
3.68/1.83	
3.68/1.83	      0: f5 -> f5 : A'=-1+A, [ A>=1 ], cost: 1
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	   Accelerated rule 0 with metering function A, yielding the new rule 4.
3.68/1.83	
3.68/1.83	   Removing the simple loops: 0.
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Accelerated all simple loops using metering functions (where possible):
3.68/1.83	
3.68/1.83	   Start location: f300
3.68/1.83	
3.68/1.83	      4: f5 -> f5 : A'=0, [ A>=1 ], cost: A
3.68/1.83	
3.68/1.83	      2: f300 -> f5 : A'=-1+A, [ A>=1 ], cost: 1
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Chained accelerated rules (with incoming rules):
3.68/1.83	
3.68/1.83	   Start location: f300
3.68/1.83	
3.68/1.83	      2: f300 -> f5 : A'=-1+A, [ A>=1 ], cost: 1
3.68/1.83	
3.68/1.83	      5: f300 -> f5 : A'=0, [ -1+A>=1 ], cost: A
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Removed unreachable locations (and leaf rules with constant cost):
3.68/1.83	
3.68/1.83	   Start location: f300
3.68/1.83	
3.68/1.83	      5: f300 -> f5 : A'=0, [ -1+A>=1 ], cost: A
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	### Computing asymptotic complexity ###
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Fully simplified ITS problem
3.68/1.83	
3.68/1.83	   Start location: f300
3.68/1.83	
3.68/1.83	      5: f300 -> f5 : A'=0, [ -1+A>=1 ], cost: A
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Computing asymptotic complexity for rule 5
3.68/1.83	
3.68/1.83	   Solved the limit problem by the following transformations:
3.68/1.83	
3.68/1.83	      Created initial limit problem:
3.68/1.83	
3.68/1.83	      -1+A (+/+!), A (+) [not solved]
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	      removing all constraints (solved by SMT)
3.68/1.83	
3.68/1.83	      resulting limit problem: [solved]
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	      applying transformation rule (C) using substitution {A==n}
3.68/1.83	
3.68/1.83	      resulting limit problem:
3.68/1.83	
3.68/1.83	      [solved]
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	   Solution:
3.68/1.83	
3.68/1.83	      A / n
3.68/1.83	
3.68/1.83	   Resulting cost n has complexity: Poly(n^1)
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Found new complexity Poly(n^1).
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	Obtained the following overall complexity (w.r.t. the length of the input n):
3.68/1.83	
3.68/1.83	   Complexity:  Poly(n^1)
3.68/1.83	
3.68/1.83	   Cpx degree:  1
3.68/1.83	
3.68/1.83	   Solved cost: n
3.68/1.83	
3.68/1.83	   Rule cost:   A
3.68/1.83	
3.68/1.83	   Rule guard:  [ -1+A>=1 ]
3.68/1.83	
3.68/1.83	
3.68/1.83	
3.68/1.83	WORST_CASE(Omega(n^1),?)
3.68/1.83	
3.68/1.83	
3.68/1.83	----------------------------------------
3.68/1.83	
3.68/1.83	(4)
3.68/1.83	BOUNDS(n^1, INF)
3.80/1.85	EOF