3.76/2.72	WORST_CASE(Omega(n^1), O(n^1))
3.76/2.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.76/2.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.76/2.73	
3.76/2.73	
3.76/2.73	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + -1 * Arg_1)).
3.76/2.73	
3.76/2.73	(0) CpxIntTrs
3.76/2.73	(1) Koat2 Proof [FINISHED, 62 ms]
3.76/2.73	(2) BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
3.76/2.73	(3) Loat Proof [FINISHED, 174 ms]
3.76/2.73	(4) BOUNDS(n^1, INF)
3.76/2.73	
3.76/2.73	
3.76/2.73	----------------------------------------
3.76/2.73	
3.76/2.73	(0)
3.76/2.73	Obligation:
3.76/2.73	Complexity Int TRS consisting of the following rules:
3.76/2.73	eval(A, B) -> Com_1(eval(A - 1, B + 1)) :|: A >= B + 1
3.76/2.73	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
3.76/2.73	
3.76/2.73	The start-symbols are:[start_2]
3.76/2.73	
3.76/2.73	
3.76/2.73	----------------------------------------
3.76/2.73	
3.76/2.73	(1) Koat2 Proof (FINISHED)
3.76/2.73	YES( ?, max([1, 1+Arg_0-Arg_1]) {O(n)})
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	Initial Complexity Problem:
3.76/2.73	
3.76/2.73	  Start:  start  
3.76/2.73	
3.76/2.73	  Program_Vars:  Arg_0, Arg_1  
3.76/2.73	
3.76/2.73	  Temp_Vars:    
3.76/2.73	
3.76/2.73	  Locations:  eval, start  
3.76/2.73	
3.76/2.73	  Transitions:
3.76/2.73	
3.76/2.73	  eval(Arg_0,Arg_1) -> eval(Arg_0-1,Arg_1+1):|:Arg_1+1 <= Arg_0
3.76/2.73	
3.76/2.73	  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	Timebounds: 
3.76/2.73	
3.76/2.73	  Overall timebound: max([1, 1+Arg_0-Arg_1]) {O(n)}
3.76/2.73	
3.76/2.73	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
3.76/2.73	
3.76/2.73	  1: start->eval: 1 {O(1)}
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	Costbounds:
3.76/2.73	
3.76/2.73	  Overall costbound: max([1, 1+Arg_0-Arg_1]) {O(n)}
3.76/2.73	
3.76/2.73	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
3.76/2.73	
3.76/2.73	  1: start->eval: 1 {O(1)}
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	Sizebounds:
3.76/2.73	
3.76/2.73	`Lower:
3.76/2.73	
3.76/2.73	  0: eval->eval, Arg_0: Arg_0-max([0, Arg_0-Arg_1]) {O(n)}
3.76/2.73	
3.76/2.73	  0: eval->eval, Arg_1: Arg_1 {O(n)}
3.76/2.73	
3.76/2.73	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.76/2.73	
3.76/2.73	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.76/2.73	
3.76/2.73	`Upper:
3.76/2.73	
3.76/2.73	  0: eval->eval, Arg_0: Arg_0 {O(n)}
3.76/2.73	
3.76/2.73	  0: eval->eval, Arg_1: Arg_1+max([0, Arg_0-Arg_1]) {O(n)}
3.76/2.73	
3.76/2.73	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.76/2.73	
3.76/2.73	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.76/2.73	
3.76/2.73	
3.76/2.73	----------------------------------------
3.76/2.73	
3.76/2.73	(2)
3.76/2.73	BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
3.76/2.73	
3.76/2.73	----------------------------------------
3.76/2.73	
3.76/2.73	(3) Loat Proof (FINISHED)
3.76/2.73	
3.76/2.73	
3.76/2.73	### Pre-processing the ITS problem ###
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	Initial linear ITS problem
3.76/2.73	
3.76/2.73	   Start location: start
3.76/2.73	
3.76/2.73	      0: eval -> eval : A'=-1+A, B'=1+B, [ A>=1+B ], cost: 1
3.76/2.73	
3.76/2.73	      1: start -> eval : [], cost: 1
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	### Simplification by acceleration and chaining ###
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	Accelerating simple loops of location 0.
3.76/2.73	
3.76/2.73	   Accelerating the following rules:
3.76/2.73	
3.76/2.73	      0: eval -> eval : A'=-1+A, B'=1+B, [ A>=1+B ], cost: 1
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	   Accelerated rule 0 with metering function meter (where 2*meter==A-B), yielding the new rule 2.
3.76/2.73	
3.76/2.73	   Removing the simple loops: 0.
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	Accelerated all simple loops using metering functions (where possible):
3.76/2.73	
3.76/2.73	   Start location: start
3.76/2.73	
3.76/2.73	      2: eval -> eval : A'=A-meter, B'=meter+B, [ A>=1+B && 2*meter==A-B && meter>=1 ], cost: meter
3.76/2.73	
3.76/2.73	      1: start -> eval : [], cost: 1
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	Chained accelerated rules (with incoming rules):
3.76/2.73	
3.76/2.73	   Start location: start
3.76/2.73	
3.76/2.73	      1: start -> eval : [], cost: 1
3.76/2.73	
3.76/2.73	      3: start -> eval : A'=A-meter, B'=meter+B, [ A>=1+B && 2*meter==A-B && meter>=1 ], cost: 1+meter
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	Removed unreachable locations (and leaf rules with constant cost):
3.76/2.73	
3.76/2.73	   Start location: start
3.76/2.73	
3.76/2.73	      3: start -> eval : A'=A-meter, B'=meter+B, [ A>=1+B && 2*meter==A-B && meter>=1 ], cost: 1+meter
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	### Computing asymptotic complexity ###
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	Fully simplified ITS problem
3.76/2.73	
3.76/2.73	   Start location: start
3.76/2.73	
3.76/2.73	      3: start -> eval : A'=A-meter, B'=meter+B, [ A>=1+B && 2*meter==A-B && meter>=1 ], cost: 1+meter
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	Computing asymptotic complexity for rule 3
3.76/2.73	
3.76/2.73	   Solved the limit problem by the following transformations:
3.76/2.73	
3.76/2.73	      Created initial limit problem:
3.76/2.73	
3.76/2.73	      1-A+2*meter+B (+/+!), 1+A-2*meter-B (+/+!), 1+meter (+), A-B (+/+!) [not solved]
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	      applying transformation rule (C) using substitution {A==2*meter+B}
3.76/2.73	
3.76/2.73	      resulting limit problem:
3.76/2.73	
3.76/2.73	      1 (+/+!), 1+meter (+), 2*meter (+/+!) [not solved]
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	      applying transformation rule (B), deleting 1 (+/+!)
3.76/2.73	
3.76/2.73	      resulting limit problem:
3.76/2.73	
3.76/2.73	      1+meter (+), 2*meter (+/+!) [not solved]
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	      removing all constraints (solved by SMT)
3.76/2.73	
3.76/2.73	      resulting limit problem: [solved]
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	      applying transformation rule (C) using substitution {meter==n}
3.76/2.73	
3.76/2.73	      resulting limit problem:
3.76/2.73	
3.76/2.73	      [solved]
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	   Solved the limit problem by the following transformations:
3.76/2.73	
3.76/2.73	      Created initial limit problem:
3.76/2.73	
3.76/2.73	      1-A+2*meter+B (+/+!), 1+A-2*meter-B (+/+!), 1+meter (+), A-B (+/+!) [not solved]
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	      applying transformation rule (C) using substitution {A==2*meter+B}
3.76/2.73	
3.76/2.73	      resulting limit problem:
3.76/2.73	
3.76/2.73	      1 (+/+!), 1+meter (+), 2*meter (+/+!) [not solved]
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	      applying transformation rule (B), deleting 1 (+/+!)
3.76/2.73	
3.76/2.73	      resulting limit problem:
3.76/2.73	
3.76/2.73	      1+meter (+), 2*meter (+/+!) [not solved]
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	      removing all constraints (solved by SMT)
3.76/2.73	
3.76/2.73	      resulting limit problem: [solved]
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	      applying transformation rule (C) using substitution {meter==n}
3.76/2.73	
3.76/2.73	      resulting limit problem:
3.76/2.73	
3.76/2.73	      [solved]
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	   Solution:
3.76/2.73	
3.76/2.73	      A / 2*n
3.76/2.73	
3.76/2.73	      meter / n
3.76/2.73	
3.76/2.73	      B / 0
3.76/2.73	
3.76/2.73	   Resulting cost 1+n has complexity: Poly(n^1)
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	Found new complexity Poly(n^1).
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	Obtained the following overall complexity (w.r.t. the length of the input n):
3.76/2.73	
3.76/2.73	   Complexity:  Poly(n^1)
3.76/2.73	
3.76/2.73	   Cpx degree:  1
3.76/2.73	
3.76/2.73	   Solved cost: 1+n
3.76/2.73	
3.76/2.73	   Rule cost:   1+meter
3.76/2.73	
3.76/2.73	   Rule guard:  [ A>=1+B && 2*meter==A-B ]
3.76/2.73	
3.76/2.73	
3.76/2.73	
3.76/2.73	WORST_CASE(Omega(n^1),?)
3.76/2.73	
3.76/2.73	
3.76/2.73	----------------------------------------
3.76/2.73	
3.76/2.73	(4)
3.76/2.73	BOUNDS(n^1, INF)
3.76/2.75	EOF