3.72/1.77	WORST_CASE(Omega(n^1), O(n^1))
3.72/1.78	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
3.72/1.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.72/1.78	
3.72/1.78	
3.72/1.78	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + -1 * Arg_1)).
3.72/1.78	
3.72/1.78	(0) CpxIntTrs
3.72/1.78	(1) Koat2 Proof [FINISHED, 27 ms]
3.72/1.78	(2) BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
3.72/1.78	(3) Loat Proof [FINISHED, 114 ms]
3.72/1.78	(4) BOUNDS(n^1, INF)
3.72/1.78	
3.72/1.78	
3.72/1.78	----------------------------------------
3.72/1.78	
3.72/1.78	(0)
3.72/1.78	Obligation:
3.72/1.78	Complexity Int TRS consisting of the following rules:
3.72/1.78	eval(A, B) -> Com_1(eval(A, B + 1)) :|: A >= B + 1
3.72/1.78	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
3.72/1.78	
3.72/1.78	The start-symbols are:[start_2]
3.72/1.78	
3.72/1.78	
3.72/1.78	----------------------------------------
3.72/1.78	
3.72/1.78	(1) Koat2 Proof (FINISHED)
3.72/1.78	YES( ?, max([1, 1+Arg_0-Arg_1]) {O(n)})
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	Initial Complexity Problem:
3.72/1.78	
3.72/1.78	  Start:  start  
3.72/1.78	
3.72/1.78	  Program_Vars:  Arg_0, Arg_1  
3.72/1.78	
3.72/1.78	  Temp_Vars:    
3.72/1.78	
3.72/1.78	  Locations:  eval, start  
3.72/1.78	
3.72/1.78	  Transitions:
3.72/1.78	
3.72/1.78	  eval(Arg_0,Arg_1) -> eval(Arg_0,Arg_1+1):|:Arg_1+1 <= Arg_0
3.72/1.78	
3.72/1.78	  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	Timebounds: 
3.72/1.78	
3.72/1.78	  Overall timebound: max([1, 1+Arg_0-Arg_1]) {O(n)}
3.72/1.78	
3.72/1.78	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
3.72/1.78	
3.72/1.78	  1: start->eval: 1 {O(1)}
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	Costbounds:
3.72/1.78	
3.72/1.78	  Overall costbound: max([1, 1+Arg_0-Arg_1]) {O(n)}
3.72/1.78	
3.72/1.78	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
3.72/1.78	
3.72/1.78	  1: start->eval: 1 {O(1)}
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	Sizebounds:
3.72/1.78	
3.72/1.78	`Lower:
3.72/1.78	
3.72/1.78	  0: eval->eval, Arg_0: Arg_0 {O(n)}
3.72/1.78	
3.72/1.78	  0: eval->eval, Arg_1: Arg_1 {O(n)}
3.72/1.78	
3.72/1.78	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.72/1.78	
3.72/1.78	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.72/1.78	
3.72/1.78	`Upper:
3.72/1.78	
3.72/1.78	  0: eval->eval, Arg_0: Arg_0 {O(n)}
3.72/1.78	
3.72/1.78	  0: eval->eval, Arg_1: Arg_1+max([0, Arg_0-Arg_1]) {O(n)}
3.72/1.78	
3.72/1.78	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.72/1.78	
3.72/1.78	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.72/1.78	
3.72/1.78	
3.72/1.78	----------------------------------------
3.72/1.78	
3.72/1.78	(2)
3.72/1.78	BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
3.72/1.78	
3.72/1.78	----------------------------------------
3.72/1.78	
3.72/1.78	(3) Loat Proof (FINISHED)
3.72/1.78	
3.72/1.78	
3.72/1.78	### Pre-processing the ITS problem ###
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	Initial linear ITS problem
3.72/1.78	
3.72/1.78	   Start location: start
3.72/1.78	
3.72/1.78	      0: eval -> eval : B'=1+B, [ A>=1+B ], cost: 1
3.72/1.78	
3.72/1.78	      1: start -> eval : [], cost: 1
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	### Simplification by acceleration and chaining ###
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	Accelerating simple loops of location 0.
3.72/1.78	
3.72/1.78	   Accelerating the following rules:
3.72/1.78	
3.72/1.78	      0: eval -> eval : B'=1+B, [ A>=1+B ], cost: 1
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	   Accelerated rule 0 with metering function A-B, yielding the new rule 2.
3.72/1.78	
3.72/1.78	   Removing the simple loops: 0.
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	Accelerated all simple loops using metering functions (where possible):
3.72/1.78	
3.72/1.78	   Start location: start
3.72/1.78	
3.72/1.78	      2: eval -> eval : B'=A, [ A>=1+B ], cost: A-B
3.72/1.78	
3.72/1.78	      1: start -> eval : [], cost: 1
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	Chained accelerated rules (with incoming rules):
3.72/1.78	
3.72/1.78	   Start location: start
3.72/1.78	
3.72/1.78	      1: start -> eval : [], cost: 1
3.72/1.78	
3.72/1.78	      3: start -> eval : B'=A, [ A>=1+B ], cost: 1+A-B
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	Removed unreachable locations (and leaf rules with constant cost):
3.72/1.78	
3.72/1.78	   Start location: start
3.72/1.78	
3.72/1.78	      3: start -> eval : B'=A, [ A>=1+B ], cost: 1+A-B
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	### Computing asymptotic complexity ###
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	Fully simplified ITS problem
3.72/1.78	
3.72/1.78	   Start location: start
3.72/1.78	
3.72/1.78	      3: start -> eval : B'=A, [ A>=1+B ], cost: 1+A-B
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	Computing asymptotic complexity for rule 3
3.72/1.78	
3.72/1.78	   Solved the limit problem by the following transformations:
3.72/1.78	
3.72/1.78	      Created initial limit problem:
3.72/1.78	
3.72/1.78	      1+A-B (+), A-B (+/+!) [not solved]
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	      removing all constraints (solved by SMT)
3.72/1.78	
3.72/1.78	      resulting limit problem: [solved]
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	      applying transformation rule (C) using substitution {A==0,B==-n}
3.72/1.78	
3.72/1.78	      resulting limit problem:
3.72/1.78	
3.72/1.78	      [solved]
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	   Solution:
3.72/1.78	
3.72/1.78	      A / 0
3.72/1.78	
3.72/1.78	      B / -n
3.72/1.78	
3.72/1.78	   Resulting cost 1+n has complexity: Poly(n^1)
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	Found new complexity Poly(n^1).
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	Obtained the following overall complexity (w.r.t. the length of the input n):
3.72/1.78	
3.72/1.78	   Complexity:  Poly(n^1)
3.72/1.78	
3.72/1.78	   Cpx degree:  1
3.72/1.78	
3.72/1.78	   Solved cost: 1+n
3.72/1.78	
3.72/1.78	   Rule cost:   1+A-B
3.72/1.78	
3.72/1.78	   Rule guard:  [ A>=1+B ]
3.72/1.78	
3.72/1.78	
3.72/1.78	
3.72/1.78	WORST_CASE(Omega(n^1),?)
3.72/1.78	
3.72/1.78	
3.72/1.78	----------------------------------------
3.72/1.78	
3.72/1.78	(4)
3.72/1.78	BOUNDS(n^1, INF)
3.72/1.80	EOF