4.02/1.83	WORST_CASE(Omega(n^1), O(n^1))
4.02/1.84	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.02/1.84	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.02/1.84	
4.02/1.84	
4.02/1.84	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + -1 * Arg_1 + -1 * Arg_2)).
4.02/1.84	
4.02/1.84	(0) CpxIntTrs
4.02/1.84	(1) Koat2 Proof [FINISHED, 147 ms]
4.02/1.84	(2) BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1 + -1 * Arg_2))
4.02/1.84	(3) Loat Proof [FINISHED, 121 ms]
4.02/1.84	(4) BOUNDS(n^1, INF)
4.02/1.84	
4.02/1.84	
4.02/1.84	----------------------------------------
4.02/1.84	
4.02/1.84	(0)
4.02/1.84	Obligation:
4.02/1.84	Complexity Int TRS consisting of the following rules:
4.02/1.84	eval(A, B, C) -> Com_1(eval(A, B + 1, C + 1)) :|: A >= B + C + 1
4.02/1.84	start(A, B, C) -> Com_1(eval(A, B, C)) :|: TRUE
4.02/1.84	
4.02/1.84	The start-symbols are:[start_3]
4.02/1.84	
4.02/1.84	
4.02/1.84	----------------------------------------
4.02/1.84	
4.02/1.84	(1) Koat2 Proof (FINISHED)
4.02/1.84	YES( ?, max([1, 1+Arg_0+-(Arg_2)-Arg_1]) {O(n)})
4.02/1.84	
4.02/1.84	
4.02/1.84	
4.02/1.84	Initial Complexity Problem:
4.02/1.84	
4.02/1.84	  Start:  start  
4.02/1.84	
4.02/1.84	  Program_Vars:  Arg_0, Arg_1, Arg_2  
4.02/1.84	
4.02/1.84	  Temp_Vars:    
4.02/1.84	
4.02/1.84	  Locations:  eval, start  
4.02/1.84	
4.02/1.84	  Transitions:
4.02/1.84	
4.02/1.84	  eval(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1+1,Arg_2+1):|:Arg_1+Arg_2+1 <= Arg_0
4.02/1.85	
4.02/1.85	  start(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1,Arg_2):|:  
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	Timebounds: 
4.02/1.85	
4.02/1.85	  Overall timebound: max([1, 1+Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.02/1.85	
4.02/1.85	  0: eval->eval: max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.02/1.85	
4.02/1.85	  1: start->eval: 1 {O(1)}
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	Costbounds:
4.02/1.85	
4.02/1.85	  Overall costbound: max([1, 1+Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.02/1.85	
4.02/1.85	  0: eval->eval: max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.02/1.85	
4.02/1.85	  1: start->eval: 1 {O(1)}
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	Sizebounds:
4.02/1.85	
4.02/1.85	`Lower:
4.02/1.85	
4.02/1.85	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.02/1.85	
4.02/1.85	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.02/1.85	
4.02/1.85	  0: eval->eval, Arg_2: Arg_2 {O(n)}
4.02/1.85	
4.02/1.85	  1: start->eval, Arg_0: Arg_0 {O(n)}
4.02/1.85	
4.02/1.85	  1: start->eval, Arg_1: Arg_1 {O(n)}
4.02/1.85	
4.02/1.85	  1: start->eval, Arg_2: Arg_2 {O(n)}
4.02/1.85	
4.02/1.85	`Upper:
4.02/1.85	
4.02/1.85	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.02/1.85	
4.02/1.85	  0: eval->eval, Arg_1: Arg_1+max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.02/1.85	
4.02/1.85	  0: eval->eval, Arg_2: Arg_2+max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.02/1.85	
4.02/1.85	  1: start->eval, Arg_0: Arg_0 {O(n)}
4.02/1.85	
4.02/1.85	  1: start->eval, Arg_1: Arg_1 {O(n)}
4.02/1.85	
4.02/1.85	  1: start->eval, Arg_2: Arg_2 {O(n)}
4.02/1.85	
4.02/1.85	
4.02/1.85	----------------------------------------
4.02/1.85	
4.02/1.85	(2)
4.02/1.85	BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1 + -1 * Arg_2))
4.02/1.85	
4.02/1.85	----------------------------------------
4.02/1.85	
4.02/1.85	(3) Loat Proof (FINISHED)
4.02/1.85	
4.02/1.85	
4.02/1.85	### Pre-processing the ITS problem ###
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	Initial linear ITS problem
4.02/1.85	
4.02/1.85	   Start location: start
4.02/1.85	
4.02/1.85	      0: eval -> eval : B'=1+B, C'=1+C, [ A>=1+C+B ], cost: 1
4.02/1.85	
4.02/1.85	      1: start -> eval : [], cost: 1
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	### Simplification by acceleration and chaining ###
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	Accelerating simple loops of location 0.
4.02/1.85	
4.02/1.85	   Accelerating the following rules:
4.02/1.85	
4.02/1.85	      0: eval -> eval : B'=1+B, C'=1+C, [ A>=1+C+B ], cost: 1
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	   Accelerated rule 0 with metering function meter (where 2*meter==-C+A-B), yielding the new rule 2.
4.02/1.85	
4.02/1.85	   Removing the simple loops: 0.
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	Accelerated all simple loops using metering functions (where possible):
4.02/1.85	
4.02/1.85	   Start location: start
4.02/1.85	
4.02/1.85	      2: eval -> eval : B'=meter+B, C'=meter+C, [ A>=1+C+B && 2*meter==-C+A-B && meter>=1 ], cost: meter
4.02/1.85	
4.02/1.85	      1: start -> eval : [], cost: 1
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	Chained accelerated rules (with incoming rules):
4.02/1.85	
4.02/1.85	   Start location: start
4.02/1.85	
4.02/1.85	      1: start -> eval : [], cost: 1
4.02/1.85	
4.02/1.85	      3: start -> eval : B'=meter+B, C'=meter+C, [ A>=1+C+B && 2*meter==-C+A-B && meter>=1 ], cost: 1+meter
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	Removed unreachable locations (and leaf rules with constant cost):
4.02/1.85	
4.02/1.85	   Start location: start
4.02/1.85	
4.02/1.85	      3: start -> eval : B'=meter+B, C'=meter+C, [ A>=1+C+B && 2*meter==-C+A-B && meter>=1 ], cost: 1+meter
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	### Computing asymptotic complexity ###
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	Fully simplified ITS problem
4.02/1.85	
4.02/1.85	   Start location: start
4.02/1.85	
4.02/1.85	      3: start -> eval : B'=meter+B, C'=meter+C, [ A>=1+C+B && 2*meter==-C+A-B && meter>=1 ], cost: 1+meter
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	Computing asymptotic complexity for rule 3
4.02/1.85	
4.02/1.85	   Solved the limit problem by the following transformations:
4.02/1.85	
4.02/1.85	      Created initial limit problem:
4.02/1.85	
4.02/1.85	      -C+A-B (+/+!), 1-2*meter-C+A-B (+/+!), 1+2*meter+C-A+B (+/+!), 1+meter (+) [not solved]
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	      applying transformation rule (C) using substitution {C==-2*meter+A-B}
4.02/1.85	
4.02/1.85	      resulting limit problem:
4.02/1.85	
4.02/1.85	      1 (+/+!), 2*meter (+/+!), 1+meter (+) [not solved]
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	      applying transformation rule (B), deleting 1 (+/+!)
4.02/1.85	
4.02/1.85	      resulting limit problem:
4.02/1.85	
4.02/1.85	      2*meter (+/+!), 1+meter (+) [not solved]
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	      removing all constraints (solved by SMT)
4.02/1.85	
4.02/1.85	      resulting limit problem: [solved]
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	      applying transformation rule (C) using substitution {meter==n}
4.02/1.85	
4.02/1.85	      resulting limit problem:
4.02/1.85	
4.02/1.85	      [solved]
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	   Solved the limit problem by the following transformations:
4.02/1.85	
4.02/1.85	      Created initial limit problem:
4.02/1.85	
4.02/1.85	      -C+A-B (+/+!), 1-2*meter-C+A-B (+/+!), 1+2*meter+C-A+B (+/+!), 1+meter (+) [not solved]
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	      applying transformation rule (C) using substitution {C==-2*meter+A-B}
4.02/1.85	
4.02/1.85	      resulting limit problem:
4.02/1.85	
4.02/1.85	      1 (+/+!), 2*meter (+/+!), 1+meter (+) [not solved]
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	      applying transformation rule (B), deleting 1 (+/+!)
4.02/1.85	
4.02/1.85	      resulting limit problem:
4.02/1.85	
4.02/1.85	      2*meter (+/+!), 1+meter (+) [not solved]
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	      removing all constraints (solved by SMT)
4.02/1.85	
4.02/1.85	      resulting limit problem: [solved]
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	      applying transformation rule (C) using substitution {meter==n}
4.02/1.85	
4.02/1.85	      resulting limit problem:
4.02/1.85	
4.02/1.85	      [solved]
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	   Solution:
4.02/1.85	
4.02/1.85	      meter / n
4.02/1.85	
4.02/1.85	      C / -2*n
4.02/1.85	
4.02/1.85	      A / 0
4.02/1.85	
4.02/1.85	      B / 0
4.02/1.85	
4.02/1.85	   Resulting cost 1+n has complexity: Poly(n^1)
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	Found new complexity Poly(n^1).
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	Obtained the following overall complexity (w.r.t. the length of the input n):
4.02/1.85	
4.02/1.85	   Complexity:  Poly(n^1)
4.02/1.85	
4.02/1.85	   Cpx degree:  1
4.02/1.85	
4.02/1.85	   Solved cost: 1+n
4.02/1.85	
4.02/1.85	   Rule cost:   1+meter
4.02/1.85	
4.02/1.85	   Rule guard:  [ A>=1+C+B && 2*meter==-C+A-B ]
4.02/1.85	
4.02/1.85	
4.02/1.85	
4.02/1.85	WORST_CASE(Omega(n^1),?)
4.02/1.85	
4.02/1.85	
4.02/1.85	----------------------------------------
4.02/1.85	
4.02/1.85	(4)
4.02/1.85	BOUNDS(n^1, INF)
4.02/1.86	EOF