3.89/1.85	WORST_CASE(Omega(n^1), O(n^1))
3.89/1.86	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.89/1.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.89/1.86	
3.89/1.86	
3.89/1.86	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + -1 * Arg_1)).
3.89/1.86	
3.89/1.86	(0) CpxIntTrs
3.89/1.86	(1) Koat2 Proof [FINISHED, 29 ms]
3.89/1.86	(2) BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
3.89/1.86	(3) Loat Proof [FINISHED, 225 ms]
3.89/1.86	(4) BOUNDS(n^1, INF)
3.89/1.86	
3.89/1.86	
3.89/1.86	----------------------------------------
3.89/1.86	
3.89/1.86	(0)
3.89/1.86	Obligation:
3.89/1.86	Complexity Int TRS consisting of the following rules:
3.89/1.86	eval(A, B) -> Com_1(eval(A - C, B + D)) :|: A >= B + 1 && C >= 0 && D >= 1
3.89/1.86	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
3.89/1.86	
3.89/1.86	The start-symbols are:[start_2]
3.89/1.86	
3.89/1.86	
3.89/1.86	----------------------------------------
3.89/1.86	
3.89/1.86	(1) Koat2 Proof (FINISHED)
3.89/1.86	YES( ?, max([1, 1+Arg_0-Arg_1]) {O(n)})
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	Initial Complexity Problem:
3.89/1.86	
3.89/1.86	  Start:  start  
3.89/1.86	
3.89/1.86	  Program_Vars:  Arg_0, Arg_1  
3.89/1.86	
3.89/1.86	  Temp_Vars:  C, D  
3.89/1.86	
3.89/1.86	  Locations:  eval, start  
3.89/1.86	
3.89/1.86	  Transitions:
3.89/1.86	
3.89/1.86	  eval(Arg_0,Arg_1) -> eval(Arg_0-C,Arg_1+D):|:Arg_1+1 <= Arg_0 && 0 <= C && 1 <= D
3.89/1.86	
3.89/1.86	  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	Timebounds: 
3.89/1.86	
3.89/1.86	  Overall timebound: max([1, 1+Arg_0-Arg_1]) {O(n)}
3.89/1.86	
3.89/1.86	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
3.89/1.86	
3.89/1.86	  1: start->eval: 1 {O(1)}
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	Costbounds:
3.89/1.86	
3.89/1.86	  Overall costbound: max([1, 1+Arg_0-Arg_1]) {O(n)}
3.89/1.86	
3.89/1.86	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
3.89/1.86	
3.89/1.86	  1: start->eval: 1 {O(1)}
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	Sizebounds:
3.89/1.86	
3.89/1.86	`Lower:
3.89/1.86	
3.89/1.86	  0: eval->eval, Arg_1: Arg_1 {O(n)}
3.89/1.86	
3.89/1.86	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.89/1.86	
3.89/1.86	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.89/1.86	
3.89/1.86	`Upper:
3.89/1.86	
3.89/1.86	  0: eval->eval, Arg_0: Arg_0 {O(n)}
3.89/1.86	
3.89/1.86	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.89/1.86	
3.89/1.86	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.89/1.86	
3.89/1.86	
3.89/1.86	----------------------------------------
3.89/1.86	
3.89/1.86	(2)
3.89/1.86	BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
3.89/1.86	
3.89/1.86	----------------------------------------
3.89/1.86	
3.89/1.86	(3) Loat Proof (FINISHED)
3.89/1.86	
3.89/1.86	
3.89/1.86	### Pre-processing the ITS problem ###
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	Initial linear ITS problem
3.89/1.86	
3.89/1.86	   Start location: start
3.89/1.86	
3.89/1.86	      0: eval -> eval : A'=A-free, B'=free_1+B, [ A>=1+B && free>=0 && free_1>=1 ], cost: 1
3.89/1.86	
3.89/1.86	      1: start -> eval : [], cost: 1
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	### Simplification by acceleration and chaining ###
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	Accelerating simple loops of location 0.
3.89/1.86	
3.89/1.86	   Accelerating the following rules:
3.89/1.86	
3.89/1.86	      0: eval -> eval : A'=A-free, B'=free_1+B, [ A>=1+B && free>=0 && free_1>=1 ], cost: 1
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	      During metering: Instantiating temporary variables by {free_1==1,free==0}
3.89/1.86	
3.89/1.86	   Accelerated rule 0 with metering function A-B, yielding the new rule 2.
3.89/1.86	
3.89/1.86	   Removing the simple loops: 0.
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	Accelerated all simple loops using metering functions (where possible):
3.89/1.86	
3.89/1.86	   Start location: start
3.89/1.86	
3.89/1.86	      2: eval -> eval : A'=A, B'=A, [ A>=1+B ], cost: A-B
3.89/1.86	
3.89/1.86	      1: start -> eval : [], cost: 1
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	Chained accelerated rules (with incoming rules):
3.89/1.86	
3.89/1.86	   Start location: start
3.89/1.86	
3.89/1.86	      1: start -> eval : [], cost: 1
3.89/1.86	
3.89/1.86	      3: start -> eval : B'=A, [ A>=1+B ], cost: 1+A-B
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	Removed unreachable locations (and leaf rules with constant cost):
3.89/1.86	
3.89/1.86	   Start location: start
3.89/1.86	
3.89/1.86	      3: start -> eval : B'=A, [ A>=1+B ], cost: 1+A-B
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	### Computing asymptotic complexity ###
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	Fully simplified ITS problem
3.89/1.86	
3.89/1.86	   Start location: start
3.89/1.86	
3.89/1.86	      3: start -> eval : B'=A, [ A>=1+B ], cost: 1+A-B
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	Computing asymptotic complexity for rule 3
3.89/1.86	
3.89/1.86	   Solved the limit problem by the following transformations:
3.89/1.86	
3.89/1.86	      Created initial limit problem:
3.89/1.86	
3.89/1.86	      1+A-B (+), A-B (+/+!) [not solved]
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	      removing all constraints (solved by SMT)
3.89/1.86	
3.89/1.86	      resulting limit problem: [solved]
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	      applying transformation rule (C) using substitution {A==0,B==-n}
3.89/1.86	
3.89/1.86	      resulting limit problem:
3.89/1.86	
3.89/1.86	      [solved]
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	   Solution:
3.89/1.86	
3.89/1.86	      A / 0
3.89/1.86	
3.89/1.86	      B / -n
3.89/1.86	
3.89/1.86	   Resulting cost 1+n has complexity: Poly(n^1)
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	Found new complexity Poly(n^1).
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	Obtained the following overall complexity (w.r.t. the length of the input n):
3.89/1.86	
3.89/1.86	   Complexity:  Poly(n^1)
3.89/1.86	
3.89/1.86	   Cpx degree:  1
3.89/1.86	
3.89/1.86	   Solved cost: 1+n
3.89/1.86	
3.89/1.86	   Rule cost:   1+A-B
3.89/1.86	
3.89/1.86	   Rule guard:  [ A>=1+B ]
3.89/1.86	
3.89/1.86	
3.89/1.86	
3.89/1.86	WORST_CASE(Omega(n^1),?)
3.89/1.86	
3.89/1.86	
3.89/1.86	----------------------------------------
3.89/1.86	
3.89/1.86	(4)
3.89/1.86	BOUNDS(n^1, INF)
3.96/1.89	EOF