4.16/2.05	WORST_CASE(Omega(n^1), O(n^1))
4.16/2.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.16/2.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.16/2.06	
4.16/2.06	
4.16/2.06	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + 2 * Arg_0 + -2 * Arg_1 + -2 * Arg_2)).
4.16/2.06	
4.16/2.06	(0) CpxIntTrs
4.16/2.06	(1) Koat2 Proof [FINISHED, 249 ms]
4.16/2.06	(2) BOUNDS(1, max(1, 1 + 2 * Arg_0 + -2 * Arg_1 + -2 * Arg_2))
4.16/2.06	(3) Loat Proof [FINISHED, 336 ms]
4.16/2.06	(4) BOUNDS(n^1, INF)
4.16/2.06	
4.16/2.06	
4.16/2.06	----------------------------------------
4.16/2.06	
4.16/2.06	(0)
4.16/2.06	Obligation:
4.16/2.06	Complexity Int TRS consisting of the following rules:
4.16/2.06	f(A, B, C) -> Com_1(f(A, B + 1, C)) :|: A >= B + C + 1
4.16/2.06	f(A, B, C) -> Com_1(f(A, B, C + 1)) :|: A >= B + C + 1
4.16/2.06	start(A, B, C) -> Com_1(f(A, B, C)) :|: TRUE
4.16/2.06	
4.16/2.06	The start-symbols are:[start_3]
4.16/2.06	
4.16/2.06	
4.16/2.06	----------------------------------------
4.16/2.06	
4.16/2.06	(1) Koat2 Proof (FINISHED)
4.16/2.06	YES( ?, 1+2*max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)})
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	Initial Complexity Problem:
4.16/2.06	
4.16/2.06	  Start:  start  
4.16/2.06	
4.16/2.06	  Program_Vars:  Arg_0, Arg_1, Arg_2  
4.16/2.06	
4.16/2.06	  Temp_Vars:    
4.16/2.06	
4.16/2.06	  Locations:  f, start  
4.16/2.06	
4.16/2.06	  Transitions:
4.16/2.06	
4.16/2.06	  f(Arg_0,Arg_1,Arg_2) -> f(Arg_0,Arg_1+1,Arg_2):|:Arg_1+Arg_2+1 <= Arg_0
4.16/2.06	
4.16/2.06	  f(Arg_0,Arg_1,Arg_2) -> f(Arg_0,Arg_1,Arg_2+1):|:Arg_1+Arg_2+1 <= Arg_0
4.16/2.06	
4.16/2.06	  start(Arg_0,Arg_1,Arg_2) -> f(Arg_0,Arg_1,Arg_2):|:  
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	Timebounds: 
4.16/2.06	
4.16/2.06	  Overall timebound: 1+2*max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.16/2.06	
4.16/2.06	  0: f->f: max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.16/2.06	
4.16/2.06	  1: f->f: max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.16/2.06	
4.16/2.06	  2: start->f: 1 {O(1)}
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	Costbounds:
4.16/2.06	
4.16/2.06	  Overall costbound: 1+2*max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.16/2.06	
4.16/2.06	  0: f->f: max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.16/2.06	
4.16/2.06	  1: f->f: max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.16/2.06	
4.16/2.06	  2: start->f: 1 {O(1)}
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	Sizebounds:
4.16/2.06	
4.16/2.06	`Lower:
4.16/2.06	
4.16/2.06	  0: f->f, Arg_0: Arg_0 {O(n)}
4.16/2.06	
4.16/2.06	  0: f->f, Arg_1: Arg_1 {O(n)}
4.16/2.06	
4.16/2.06	  0: f->f, Arg_2: Arg_2 {O(n)}
4.16/2.06	
4.16/2.06	  1: f->f, Arg_0: Arg_0 {O(n)}
4.16/2.06	
4.16/2.06	  1: f->f, Arg_1: Arg_1 {O(n)}
4.16/2.06	
4.16/2.06	  1: f->f, Arg_2: Arg_2 {O(n)}
4.16/2.06	
4.16/2.06	  2: start->f, Arg_0: Arg_0 {O(n)}
4.16/2.06	
4.16/2.06	  2: start->f, Arg_1: Arg_1 {O(n)}
4.16/2.06	
4.16/2.06	  2: start->f, Arg_2: Arg_2 {O(n)}
4.16/2.06	
4.16/2.06	`Upper:
4.16/2.06	
4.16/2.06	  0: f->f, Arg_0: Arg_0 {O(n)}
4.16/2.06	
4.16/2.06	  0: f->f, Arg_1: Arg_1+max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.16/2.06	
4.16/2.06	  0: f->f, Arg_2: Arg_2+max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.16/2.06	
4.16/2.06	  1: f->f, Arg_0: Arg_0 {O(n)}
4.16/2.06	
4.16/2.06	  1: f->f, Arg_1: Arg_1+max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.16/2.06	
4.16/2.06	  1: f->f, Arg_2: Arg_2+max([0, Arg_0+-(Arg_2)-Arg_1]) {O(n)}
4.16/2.06	
4.16/2.06	  2: start->f, Arg_0: Arg_0 {O(n)}
4.16/2.06	
4.16/2.06	  2: start->f, Arg_1: Arg_1 {O(n)}
4.16/2.06	
4.16/2.06	  2: start->f, Arg_2: Arg_2 {O(n)}
4.16/2.06	
4.16/2.06	
4.16/2.06	----------------------------------------
4.16/2.06	
4.16/2.06	(2)
4.16/2.06	BOUNDS(1, max(1, 1 + 2 * Arg_0 + -2 * Arg_1 + -2 * Arg_2))
4.16/2.06	
4.16/2.06	----------------------------------------
4.16/2.06	
4.16/2.06	(3) Loat Proof (FINISHED)
4.16/2.06	
4.16/2.06	
4.16/2.06	### Pre-processing the ITS problem ###
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	Initial linear ITS problem
4.16/2.06	
4.16/2.06	   Start location: start
4.16/2.06	
4.16/2.06	      0: f -> f : B'=1+B, [ A>=1+C+B ], cost: 1
4.16/2.06	
4.16/2.06	      1: f -> f : C'=1+C, [ A>=1+C+B ], cost: 1
4.16/2.06	
4.16/2.06	      2: start -> f : [], cost: 1
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	### Simplification by acceleration and chaining ###
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	Accelerating simple loops of location 0.
4.16/2.06	
4.16/2.06	   Accelerating the following rules:
4.16/2.06	
4.16/2.06	      0: f -> f : B'=1+B, [ A>=1+C+B ], cost: 1
4.16/2.06	
4.16/2.06	      1: f -> f : C'=1+C, [ A>=1+C+B ], cost: 1
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	   Accelerated rule 0 with metering function -C+A-B, yielding the new rule 3.
4.16/2.06	
4.16/2.06	   Accelerated rule 1 with metering function -C+A-B, yielding the new rule 4.
4.16/2.06	
4.16/2.06	   Removing the simple loops: 0 1.
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	Accelerated all simple loops using metering functions (where possible):
4.16/2.06	
4.16/2.06	   Start location: start
4.16/2.06	
4.16/2.06	      3: f -> f : B'=-C+A, [ A>=1+C+B ], cost: -C+A-B
4.16/2.06	
4.16/2.06	      4: f -> f : C'=A-B, [ A>=1+C+B ], cost: -C+A-B
4.16/2.06	
4.16/2.06	      2: start -> f : [], cost: 1
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	Chained accelerated rules (with incoming rules):
4.16/2.06	
4.16/2.06	   Start location: start
4.16/2.06	
4.16/2.06	      2: start -> f : [], cost: 1
4.16/2.06	
4.16/2.06	      5: start -> f : B'=-C+A, [ A>=1+C+B ], cost: 1-C+A-B
4.16/2.06	
4.16/2.06	      6: start -> f : C'=A-B, [ A>=1+C+B ], cost: 1-C+A-B
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	Removed unreachable locations (and leaf rules with constant cost):
4.16/2.06	
4.16/2.06	   Start location: start
4.16/2.06	
4.16/2.06	      5: start -> f : B'=-C+A, [ A>=1+C+B ], cost: 1-C+A-B
4.16/2.06	
4.16/2.06	      6: start -> f : C'=A-B, [ A>=1+C+B ], cost: 1-C+A-B
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	### Computing asymptotic complexity ###
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	Fully simplified ITS problem
4.16/2.06	
4.16/2.06	   Start location: start
4.16/2.06	
4.16/2.06	      6: start -> f : C'=A-B, [ A>=1+C+B ], cost: 1-C+A-B
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	Computing asymptotic complexity for rule 6
4.16/2.06	
4.16/2.06	   Solved the limit problem by the following transformations:
4.16/2.06	
4.16/2.06	      Created initial limit problem:
4.16/2.06	
4.16/2.06	      1-C+A-B (+), -C+A-B (+/+!) [not solved]
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	      removing all constraints (solved by SMT)
4.16/2.06	
4.16/2.06	      resulting limit problem: [solved]
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	      applying transformation rule (C) using substitution {C==0,A==0,B==-n}
4.16/2.06	
4.16/2.06	      resulting limit problem:
4.16/2.06	
4.16/2.06	      [solved]
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	   Solution:
4.16/2.06	
4.16/2.06	      C / 0
4.16/2.06	
4.16/2.06	      A / 0
4.16/2.06	
4.16/2.06	      B / -n
4.16/2.06	
4.16/2.06	   Resulting cost 1+n has complexity: Poly(n^1)
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	Found new complexity Poly(n^1).
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	Obtained the following overall complexity (w.r.t. the length of the input n):
4.16/2.06	
4.16/2.06	   Complexity:  Poly(n^1)
4.16/2.06	
4.16/2.06	   Cpx degree:  1
4.16/2.06	
4.16/2.06	   Solved cost: 1+n
4.16/2.06	
4.16/2.06	   Rule cost:   1-C+A-B
4.16/2.06	
4.16/2.06	   Rule guard:  [ A>=1+C+B ]
4.16/2.06	
4.16/2.06	
4.16/2.06	
4.16/2.06	WORST_CASE(Omega(n^1),?)
4.16/2.06	
4.16/2.06	
4.16/2.06	----------------------------------------
4.16/2.06	
4.16/2.06	(4)
4.16/2.06	BOUNDS(n^1, INF)
4.16/2.09	EOF