4.11/2.26	WORST_CASE(Omega(n^1), O(n^1))
4.23/2.26	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
4.23/2.26	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.23/2.26	
4.23/2.26	
4.23/2.26	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + Arg_1)).
4.23/2.26	
4.23/2.26	(0) CpxIntTrs
4.23/2.26	(1) Koat2 Proof [FINISHED, 46 ms]
4.23/2.26	(2) BOUNDS(1, max(1, 1 + Arg_0 + Arg_1))
4.23/2.26	(3) Loat Proof [FINISHED, 630 ms]
4.23/2.26	(4) BOUNDS(n^1, INF)
4.23/2.26	
4.23/2.26	
4.23/2.26	----------------------------------------
4.23/2.26	
4.23/2.26	(0)
4.23/2.26	Obligation:
4.23/2.26	Complexity Int TRS consisting of the following rules:
4.23/2.26	f0(A, B) -> Com_1(f1(A, B)) :|: A >= 1
4.23/2.26	f1(A, B) -> Com_1(f1(A, B - A)) :|: A >= 1 && B >= 0
4.23/2.26	
4.23/2.26	The start-symbols are:[f0_2]
4.23/2.26	
4.23/2.26	
4.23/2.26	----------------------------------------
4.23/2.26	
4.23/2.26	(1) Koat2 Proof (FINISHED)
4.23/2.26	YES( ?, max([1, 1+Arg_0+Arg_1]) {O(n)})
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	Initial Complexity Problem:
4.23/2.26	
4.23/2.26	  Start:  f0  
4.23/2.26	
4.23/2.26	  Program_Vars:  Arg_0, Arg_1  
4.23/2.26	
4.23/2.26	  Temp_Vars:    
4.23/2.26	
4.23/2.26	  Locations:  f0, f1  
4.23/2.26	
4.23/2.26	  Transitions:
4.23/2.26	
4.23/2.26	  f0(Arg_0,Arg_1) -> f1(Arg_0,Arg_1):|:1 <= Arg_0
4.23/2.26	
4.23/2.26	  f1(Arg_0,Arg_1) -> f1(Arg_0,Arg_1-Arg_0):|:1 <= Arg_0 && 1 <= Arg_0 && 0 <= Arg_1  
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	Timebounds: 
4.23/2.26	
4.23/2.26	  Overall timebound: max([1, 1+Arg_0+Arg_1]) {O(n)}
4.23/2.26	
4.23/2.26	  0: f0->f1: 1 {O(1)}
4.23/2.26	
4.23/2.26	  1: f1->f1: max([0, Arg_0+Arg_1]) {O(n)}
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	Costbounds:
4.23/2.26	
4.23/2.26	  Overall costbound: max([1, 1+Arg_0+Arg_1]) {O(n)}
4.23/2.26	
4.23/2.26	  0: f0->f1: 1 {O(1)}
4.23/2.26	
4.23/2.26	  1: f1->f1: max([0, Arg_0+Arg_1]) {O(n)}
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	Sizebounds:
4.23/2.26	
4.23/2.26	`Lower:
4.23/2.26	
4.23/2.26	  0: f0->f1, Arg_0: 1 {O(1)}
4.23/2.26	
4.23/2.26	  0: f0->f1, Arg_1: Arg_1 {O(n)}
4.23/2.26	
4.23/2.26	  1: f1->f1, Arg_0: 1 {O(1)}
4.23/2.26	
4.23/2.26	  1: f1->f1, Arg_1: -(Arg_0) {O(n)}
4.23/2.26	
4.23/2.26	`Upper:
4.23/2.26	
4.23/2.26	  0: f0->f1, Arg_0: Arg_0 {O(n)}
4.23/2.26	
4.23/2.26	  0: f0->f1, Arg_1: Arg_1 {O(n)}
4.23/2.26	
4.23/2.26	  1: f1->f1, Arg_0: Arg_0 {O(n)}
4.23/2.26	
4.23/2.26	  1: f1->f1, Arg_1: Arg_1 {O(n)}
4.23/2.26	
4.23/2.26	
4.23/2.26	----------------------------------------
4.23/2.26	
4.23/2.26	(2)
4.23/2.26	BOUNDS(1, max(1, 1 + Arg_0 + Arg_1))
4.23/2.26	
4.23/2.26	----------------------------------------
4.23/2.26	
4.23/2.26	(3) Loat Proof (FINISHED)
4.23/2.26	
4.23/2.26	
4.23/2.26	### Pre-processing the ITS problem ###
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	Initial linear ITS problem
4.23/2.26	
4.23/2.26	   Start location: f0
4.23/2.26	
4.23/2.26	      0: f0 -> f1 : [ A>=1 ], cost: 1
4.23/2.26	
4.23/2.26	      1: f1 -> f1 : B'=-A+B, [ A>=1 && B>=0 ], cost: 1
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	### Simplification by acceleration and chaining ###
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	Accelerating simple loops of location 1.
4.23/2.26	
4.23/2.26	   Accelerating the following rules:
4.23/2.26	
4.23/2.26	      1: f1 -> f1 : B'=-A+B, [ A>=1 && B>=0 ], cost: 1
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	   Accelerated rule 1 with backward acceleration, yielding the new rule 2.
4.23/2.26	
4.23/2.26	   Removing the simple loops: 1.
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	Accelerated all simple loops using metering functions (where possible):
4.23/2.26	
4.23/2.26	   Start location: f0
4.23/2.26	
4.23/2.26	      0: f0 -> f1 : [ A>=1 ], cost: 1
4.23/2.26	
4.23/2.26	      2: f1 -> f1 : B'=-k*A+B, [ A>=1 && B>=0 && k>0 && -(-1+k)*A+B>=0 ], cost: k
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	Chained accelerated rules (with incoming rules):
4.23/2.26	
4.23/2.26	   Start location: f0
4.23/2.26	
4.23/2.26	      0: f0 -> f1 : [ A>=1 ], cost: 1
4.23/2.26	
4.23/2.26	      3: f0 -> f1 : B'=-k*A+B, [ A>=1 && B>=0 && k>0 && -(-1+k)*A+B>=0 ], cost: 1+k
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	Removed unreachable locations (and leaf rules with constant cost):
4.23/2.26	
4.23/2.26	   Start location: f0
4.23/2.26	
4.23/2.26	      3: f0 -> f1 : B'=-k*A+B, [ A>=1 && B>=0 && k>0 && -(-1+k)*A+B>=0 ], cost: 1+k
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	### Computing asymptotic complexity ###
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	Fully simplified ITS problem
4.23/2.26	
4.23/2.26	   Start location: f0
4.23/2.26	
4.23/2.26	      3: f0 -> f1 : B'=-k*A+B, [ A>=1 && B>=0 && k>0 && -(-1+k)*A+B>=0 ], cost: 1+k
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	Computing asymptotic complexity for rule 3
4.23/2.26	
4.23/2.26	   Solved the limit problem by the following transformations:
4.23/2.26	
4.23/2.26	      Created initial limit problem:
4.23/2.26	
4.23/2.26	      1+B (+/+!), 1-(-1+k)*A+B (+/+!), k (+/+!), 1+k (+), A (+/+!) [not solved]
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	      removing all constraints (solved by SMT)
4.23/2.26	
4.23/2.26	      resulting limit problem: [solved]
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	      applying transformation rule (C) using substitution {k==n,A==1,B==2*n}
4.23/2.26	
4.23/2.26	      resulting limit problem:
4.23/2.26	
4.23/2.26	      [solved]
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	   Solved the limit problem by the following transformations:
4.23/2.26	
4.23/2.26	      Created initial limit problem:
4.23/2.26	
4.23/2.26	      1+B (+/+!), 1-(-1+k)*A+B (+/+!), k (+/+!), 1+k (+), A (+/+!) [not solved]
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	      applying transformation rule (C) using substitution {A==1}
4.23/2.26	
4.23/2.26	      resulting limit problem:
4.23/2.26	
4.23/2.26	      1 (+/+!), 2-k+B (+/+!), 1+B (+/+!), k (+/+!), 1+k (+) [not solved]
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	      applying transformation rule (B), deleting 1 (+/+!)
4.23/2.26	
4.23/2.26	      resulting limit problem:
4.23/2.26	
4.23/2.26	      2-k+B (+/+!), 1+B (+/+!), k (+/+!), 1+k (+) [not solved]
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	      removing all constraints (solved by SMT)
4.23/2.26	
4.23/2.26	      resulting limit problem: [solved]
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	      applying transformation rule (C) using substitution {k==n,B==n}
4.23/2.26	
4.23/2.26	      resulting limit problem:
4.23/2.26	
4.23/2.26	      [solved]
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	   Solution:
4.23/2.26	
4.23/2.26	      k / n
4.23/2.26	
4.23/2.26	      A / 1
4.23/2.26	
4.23/2.26	      B / 2*n
4.23/2.26	
4.23/2.26	   Resulting cost 1+n has complexity: Poly(n^1)
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	Found new complexity Poly(n^1).
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	Obtained the following overall complexity (w.r.t. the length of the input n):
4.23/2.26	
4.23/2.26	   Complexity:  Poly(n^1)
4.23/2.26	
4.23/2.26	   Cpx degree:  1
4.23/2.26	
4.23/2.26	   Solved cost: 1+n
4.23/2.26	
4.23/2.26	   Rule cost:   1+k
4.23/2.26	
4.23/2.26	   Rule guard:  [ A>=1 && B>=0 && k>0 && -(-1+k)*A+B>=0 ]
4.23/2.26	
4.23/2.26	
4.23/2.26	
4.23/2.26	WORST_CASE(Omega(n^1),?)
4.23/2.26	
4.23/2.26	
4.23/2.26	----------------------------------------
4.23/2.26	
4.23/2.26	(4)
4.23/2.26	BOUNDS(n^1, INF)
4.23/2.29	EOF