4.02/1.86	WORST_CASE(Omega(n^1), O(n^1))
4.02/1.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.02/1.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.02/1.87	
4.02/1.87	
4.02/1.87	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + -1 * Arg_1)).
4.02/1.87	
4.02/1.87	(0) CpxIntTrs
4.02/1.87	(1) Koat2 Proof [FINISHED, 42 ms]
4.02/1.87	(2) BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
4.02/1.87	(3) Loat Proof [FINISHED, 141 ms]
4.02/1.87	(4) BOUNDS(n^1, INF)
4.02/1.87	
4.02/1.87	
4.02/1.87	----------------------------------------
4.02/1.87	
4.02/1.87	(0)
4.02/1.87	Obligation:
4.02/1.87	Complexity Int TRS consisting of the following rules:
4.02/1.87	eval(A, B) -> Com_1(eval(A + 1, B + 2)) :|: A >= B + 1
4.02/1.87	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
4.02/1.87	
4.02/1.87	The start-symbols are:[start_2]
4.02/1.87	
4.02/1.87	
4.02/1.87	----------------------------------------
4.02/1.87	
4.02/1.87	(1) Koat2 Proof (FINISHED)
4.02/1.87	YES( ?, max([1, 1+Arg_0-Arg_1]) {O(n)})
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	Initial Complexity Problem:
4.02/1.87	
4.02/1.87	  Start:  start  
4.02/1.87	
4.02/1.87	  Program_Vars:  Arg_0, Arg_1  
4.02/1.87	
4.02/1.87	  Temp_Vars:    
4.02/1.87	
4.02/1.87	  Locations:  eval, start  
4.02/1.87	
4.02/1.87	  Transitions:
4.02/1.87	
4.02/1.87	  eval(Arg_0,Arg_1) -> eval(Arg_0+1,Arg_1+2):|:Arg_1+1 <= Arg_0
4.02/1.87	
4.02/1.87	  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	Timebounds: 
4.02/1.87	
4.02/1.87	  Overall timebound: max([1, 1+Arg_0-Arg_1]) {O(n)}
4.02/1.87	
4.02/1.87	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
4.02/1.87	
4.02/1.87	  1: start->eval: 1 {O(1)}
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	Costbounds:
4.02/1.87	
4.02/1.87	  Overall costbound: max([1, 1+Arg_0-Arg_1]) {O(n)}
4.02/1.87	
4.02/1.87	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
4.02/1.87	
4.02/1.87	  1: start->eval: 1 {O(1)}
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	Sizebounds:
4.02/1.87	
4.02/1.87	`Lower:
4.02/1.87	
4.02/1.87	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.02/1.87	
4.02/1.87	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.02/1.87	
4.02/1.87	  1: start->eval, Arg_0: Arg_0 {O(n)}
4.02/1.87	
4.02/1.87	  1: start->eval, Arg_1: Arg_1 {O(n)}
4.02/1.87	
4.02/1.87	`Upper:
4.02/1.87	
4.02/1.87	  0: eval->eval, Arg_0: Arg_0+max([0, Arg_0-Arg_1]) {O(n)}
4.02/1.87	
4.02/1.87	  0: eval->eval, Arg_1: Arg_1+max([0, 2*(Arg_0-Arg_1)]) {O(n)}
4.02/1.87	
4.02/1.87	  1: start->eval, Arg_0: Arg_0 {O(n)}
4.02/1.87	
4.02/1.87	  1: start->eval, Arg_1: Arg_1 {O(n)}
4.02/1.87	
4.02/1.87	
4.02/1.87	----------------------------------------
4.02/1.87	
4.02/1.87	(2)
4.02/1.87	BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
4.02/1.87	
4.02/1.87	----------------------------------------
4.02/1.87	
4.02/1.87	(3) Loat Proof (FINISHED)
4.02/1.87	
4.02/1.87	
4.02/1.87	### Pre-processing the ITS problem ###
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	Initial linear ITS problem
4.02/1.87	
4.02/1.87	   Start location: start
4.02/1.87	
4.02/1.87	      0: eval -> eval : A'=1+A, B'=2+B, [ A>=1+B ], cost: 1
4.02/1.87	
4.02/1.87	      1: start -> eval : [], cost: 1
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	### Simplification by acceleration and chaining ###
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	Accelerating simple loops of location 0.
4.02/1.87	
4.02/1.87	   Accelerating the following rules:
4.02/1.87	
4.02/1.87	      0: eval -> eval : A'=1+A, B'=2+B, [ A>=1+B ], cost: 1
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	   Accelerated rule 0 with metering function A-B, yielding the new rule 2.
4.02/1.87	
4.02/1.87	   Removing the simple loops: 0.
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	Accelerated all simple loops using metering functions (where possible):
4.02/1.87	
4.02/1.87	   Start location: start
4.02/1.87	
4.02/1.87	      2: eval -> eval : A'=2*A-B, B'=2*A-B, [ A>=1+B ], cost: A-B
4.02/1.87	
4.02/1.87	      1: start -> eval : [], cost: 1
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	Chained accelerated rules (with incoming rules):
4.02/1.87	
4.02/1.87	   Start location: start
4.02/1.87	
4.02/1.87	      1: start -> eval : [], cost: 1
4.02/1.87	
4.02/1.87	      3: start -> eval : A'=2*A-B, B'=2*A-B, [ A>=1+B ], cost: 1+A-B
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	Removed unreachable locations (and leaf rules with constant cost):
4.02/1.87	
4.02/1.87	   Start location: start
4.02/1.87	
4.02/1.87	      3: start -> eval : A'=2*A-B, B'=2*A-B, [ A>=1+B ], cost: 1+A-B
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	### Computing asymptotic complexity ###
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	Fully simplified ITS problem
4.02/1.87	
4.02/1.87	   Start location: start
4.02/1.87	
4.02/1.87	      3: start -> eval : A'=2*A-B, B'=2*A-B, [ A>=1+B ], cost: 1+A-B
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	Computing asymptotic complexity for rule 3
4.02/1.87	
4.02/1.87	   Solved the limit problem by the following transformations:
4.02/1.87	
4.02/1.87	      Created initial limit problem:
4.02/1.87	
4.02/1.87	      1+A-B (+), A-B (+/+!) [not solved]
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	      removing all constraints (solved by SMT)
4.02/1.87	
4.02/1.87	      resulting limit problem: [solved]
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	      applying transformation rule (C) using substitution {A==0,B==-n}
4.02/1.87	
4.02/1.87	      resulting limit problem:
4.02/1.87	
4.02/1.87	      [solved]
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	   Solution:
4.02/1.87	
4.02/1.87	      A / 0
4.02/1.87	
4.02/1.87	      B / -n
4.02/1.87	
4.02/1.87	   Resulting cost 1+n has complexity: Poly(n^1)
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	Found new complexity Poly(n^1).
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	Obtained the following overall complexity (w.r.t. the length of the input n):
4.02/1.87	
4.02/1.87	   Complexity:  Poly(n^1)
4.02/1.87	
4.02/1.87	   Cpx degree:  1
4.02/1.87	
4.02/1.87	   Solved cost: 1+n
4.02/1.87	
4.02/1.87	   Rule cost:   1+A-B
4.02/1.87	
4.02/1.87	   Rule guard:  [ A>=1+B ]
4.02/1.87	
4.02/1.87	
4.02/1.87	
4.02/1.87	WORST_CASE(Omega(n^1),?)
4.02/1.87	
4.02/1.87	
4.02/1.87	----------------------------------------
4.02/1.87	
4.02/1.87	(4)
4.02/1.87	BOUNDS(n^1, INF)
4.02/1.90	EOF