3.66/1.78	WORST_CASE(Omega(n^1), O(n^1))
3.66/1.79	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.66/1.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.66/1.79	
3.66/1.79	
3.66/1.79	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + -1 * Arg_1)).
3.66/1.79	
3.66/1.79	(0) CpxIntTrs
3.66/1.79	(1) Koat2 Proof [FINISHED, 27 ms]
3.66/1.79	(2) BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
3.66/1.79	(3) Loat Proof [FINISHED, 120 ms]
3.66/1.79	(4) BOUNDS(n^1, INF)
3.66/1.79	
3.66/1.79	
3.66/1.79	----------------------------------------
3.66/1.79	
3.66/1.79	(0)
3.66/1.79	Obligation:
3.66/1.79	Complexity Int TRS consisting of the following rules:
3.66/1.79	eval(A, B) -> Com_1(eval(A - 1, B)) :|: A >= B + 1
3.66/1.79	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
3.66/1.79	
3.66/1.79	The start-symbols are:[start_2]
3.66/1.79	
3.66/1.79	
3.66/1.79	----------------------------------------
3.66/1.79	
3.66/1.79	(1) Koat2 Proof (FINISHED)
3.66/1.79	YES( ?, max([1, 1+Arg_0-Arg_1]) {O(n)})
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	Initial Complexity Problem:
3.66/1.79	
3.66/1.79	  Start:  start  
3.66/1.79	
3.66/1.79	  Program_Vars:  Arg_0, Arg_1  
3.66/1.79	
3.66/1.79	  Temp_Vars:    
3.66/1.79	
3.66/1.79	  Locations:  eval, start  
3.66/1.79	
3.66/1.79	  Transitions:
3.66/1.79	
3.66/1.79	  eval(Arg_0,Arg_1) -> eval(Arg_0-1,Arg_1):|:Arg_1+1 <= Arg_0
3.66/1.79	
3.66/1.79	  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	Timebounds: 
3.66/1.79	
3.66/1.79	  Overall timebound: max([1, 1+Arg_0-Arg_1]) {O(n)}
3.66/1.79	
3.66/1.79	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
3.66/1.79	
3.66/1.79	  1: start->eval: 1 {O(1)}
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	Costbounds:
3.66/1.79	
3.66/1.79	  Overall costbound: max([1, 1+Arg_0-Arg_1]) {O(n)}
3.66/1.79	
3.66/1.79	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
3.66/1.79	
3.66/1.79	  1: start->eval: 1 {O(1)}
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	Sizebounds:
3.66/1.79	
3.66/1.79	`Lower:
3.66/1.79	
3.66/1.79	  0: eval->eval, Arg_0: Arg_0-max([0, Arg_0-Arg_1]) {O(n)}
3.66/1.79	
3.66/1.79	  0: eval->eval, Arg_1: Arg_1 {O(n)}
3.66/1.79	
3.66/1.79	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.66/1.79	
3.66/1.79	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.66/1.79	
3.66/1.79	`Upper:
3.66/1.79	
3.66/1.79	  0: eval->eval, Arg_0: Arg_0 {O(n)}
3.66/1.79	
3.66/1.79	  0: eval->eval, Arg_1: Arg_1 {O(n)}
3.66/1.79	
3.66/1.79	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.66/1.79	
3.66/1.79	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.66/1.79	
3.66/1.79	
3.66/1.79	----------------------------------------
3.66/1.79	
3.66/1.79	(2)
3.66/1.79	BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
3.66/1.79	
3.66/1.79	----------------------------------------
3.66/1.79	
3.66/1.79	(3) Loat Proof (FINISHED)
3.66/1.79	
3.66/1.79	
3.66/1.79	### Pre-processing the ITS problem ###
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	Initial linear ITS problem
3.66/1.79	
3.66/1.79	   Start location: start
3.66/1.79	
3.66/1.79	      0: eval -> eval : A'=-1+A, [ A>=1+B ], cost: 1
3.66/1.79	
3.66/1.79	      1: start -> eval : [], cost: 1
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	### Simplification by acceleration and chaining ###
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	Accelerating simple loops of location 0.
3.66/1.79	
3.66/1.79	   Accelerating the following rules:
3.66/1.79	
3.66/1.79	      0: eval -> eval : A'=-1+A, [ A>=1+B ], cost: 1
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	   Accelerated rule 0 with metering function A-B, yielding the new rule 2.
3.66/1.79	
3.66/1.79	   Removing the simple loops: 0.
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	Accelerated all simple loops using metering functions (where possible):
3.66/1.79	
3.66/1.79	   Start location: start
3.66/1.79	
3.66/1.79	      2: eval -> eval : A'=B, [ A>=1+B ], cost: A-B
3.66/1.79	
3.66/1.79	      1: start -> eval : [], cost: 1
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	Chained accelerated rules (with incoming rules):
3.66/1.79	
3.66/1.79	   Start location: start
3.66/1.79	
3.66/1.79	      1: start -> eval : [], cost: 1
3.66/1.79	
3.66/1.79	      3: start -> eval : A'=B, [ A>=1+B ], cost: 1+A-B
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	Removed unreachable locations (and leaf rules with constant cost):
3.66/1.79	
3.66/1.79	   Start location: start
3.66/1.79	
3.66/1.79	      3: start -> eval : A'=B, [ A>=1+B ], cost: 1+A-B
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	### Computing asymptotic complexity ###
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	Fully simplified ITS problem
3.66/1.79	
3.66/1.79	   Start location: start
3.66/1.79	
3.66/1.79	      3: start -> eval : A'=B, [ A>=1+B ], cost: 1+A-B
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	Computing asymptotic complexity for rule 3
3.66/1.79	
3.66/1.79	   Solved the limit problem by the following transformations:
3.66/1.79	
3.66/1.79	      Created initial limit problem:
3.66/1.79	
3.66/1.79	      1+A-B (+), A-B (+/+!) [not solved]
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	      removing all constraints (solved by SMT)
3.66/1.79	
3.66/1.79	      resulting limit problem: [solved]
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	      applying transformation rule (C) using substitution {A==0,B==-n}
3.66/1.79	
3.66/1.79	      resulting limit problem:
3.66/1.79	
3.66/1.79	      [solved]
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	   Solution:
3.66/1.79	
3.66/1.79	      A / 0
3.66/1.79	
3.66/1.79	      B / -n
3.66/1.79	
3.66/1.79	   Resulting cost 1+n has complexity: Poly(n^1)
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	Found new complexity Poly(n^1).
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	Obtained the following overall complexity (w.r.t. the length of the input n):
3.66/1.79	
3.66/1.79	   Complexity:  Poly(n^1)
3.66/1.79	
3.66/1.79	   Cpx degree:  1
3.66/1.79	
3.66/1.79	   Solved cost: 1+n
3.66/1.79	
3.66/1.79	   Rule cost:   1+A-B
3.66/1.79	
3.66/1.79	   Rule guard:  [ A>=1+B ]
3.66/1.79	
3.66/1.79	
3.66/1.79	
3.66/1.79	WORST_CASE(Omega(n^1),?)
3.66/1.79	
3.66/1.79	
3.66/1.79	----------------------------------------
3.66/1.79	
3.66/1.79	(4)
3.66/1.79	BOUNDS(n^1, INF)
3.66/1.81	EOF