3.56/2.08	WORST_CASE(Omega(n^1), O(n^1))
4.03/2.08	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
4.03/2.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.03/2.08	
4.03/2.08	
4.03/2.08	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, nat(Arg_1) + max(1, 1 + Arg_0)).
4.03/2.08	
4.03/2.08	(0) CpxIntTrs
4.03/2.08	(1) Koat2 Proof [FINISHED, 169 ms]
4.03/2.08	(2) BOUNDS(1, nat(Arg_1) + max(1, 1 + Arg_0))
4.03/2.08	(3) Loat Proof [FINISHED, 417 ms]
4.03/2.08	(4) BOUNDS(n^1, INF)
4.03/2.08	
4.03/2.08	
4.03/2.08	----------------------------------------
4.03/2.08	
4.03/2.08	(0)
4.03/2.08	Obligation:
4.03/2.08	Complexity Int TRS consisting of the following rules:
4.03/2.08	f1(A, B) -> Com_1(f2(A, B)) :|: A >= 1 && B >= 1
4.03/2.08	f2(A, B) -> Com_1(f2(A - 1, B)) :|: A >= 2 && B >= 1
4.03/2.08	f2(A, B) -> Com_1(f2(A, B - 1)) :|: A >= 1 && B >= 2
4.03/2.08	
4.03/2.08	The start-symbols are:[f1_2]
4.03/2.08	
4.03/2.08	
4.03/2.08	----------------------------------------
4.03/2.08	
4.03/2.08	(1) Koat2 Proof (FINISHED)
4.03/2.08	YES( ?, max([0, Arg_1])+max([1, 1+Arg_0]) {O(n)})
4.03/2.08	
4.03/2.08	
4.03/2.08	
4.03/2.08	Initial Complexity Problem:
4.03/2.08	
4.03/2.08	  Start:  f1  
4.03/2.08	
4.03/2.08	  Program_Vars:  Arg_0, Arg_1  
4.03/2.08	
4.03/2.08	  Temp_Vars:    
4.03/2.08	
4.03/2.08	  Locations:  f1, f2  
4.03/2.08	
4.03/2.08	  Transitions:
4.03/2.08	
4.03/2.08	  f1(Arg_0,Arg_1) -> f2(Arg_0,Arg_1):|:1 <= Arg_0 && 1 <= Arg_1
4.03/2.08	
4.03/2.08	  f2(Arg_0,Arg_1) -> f2(Arg_0-1,Arg_1):|:1 <= Arg_1 && 2 <= Arg_0+Arg_1 && 1 <= Arg_0 && 2 <= Arg_0 && 1 <= Arg_1
4.03/2.08	
4.03/2.08	  f2(Arg_0,Arg_1) -> f2(Arg_0,Arg_1-1):|:1 <= Arg_1 && 2 <= Arg_0+Arg_1 && 1 <= Arg_0 && 1 <= Arg_0 && 2 <= Arg_1  
4.03/2.08	
4.03/2.08	
4.03/2.08	
4.03/2.08	Timebounds: 
4.03/2.08	
4.03/2.08	  Overall timebound: max([0, Arg_1])+max([1, 1+Arg_0]) {O(n)}
4.03/2.08	
4.03/2.08	  0: f1->f2: 1 {O(1)}
4.03/2.08	
4.03/2.08	  1: f2->f2: max([0, Arg_0]) {O(n)}
4.03/2.08	
4.03/2.08	  2: f2->f2: max([0, Arg_1]) {O(n)}
4.03/2.08	
4.03/2.08	
4.03/2.08	
4.03/2.08	Costbounds:
4.03/2.08	
4.03/2.08	  Overall costbound: max([0, Arg_1])+max([1, 1+Arg_0]) {O(n)}
4.03/2.08	
4.03/2.08	  0: f1->f2: 1 {O(1)}
4.03/2.08	
4.03/2.08	  1: f2->f2: max([0, Arg_0]) {O(n)}
4.03/2.08	
4.03/2.08	  2: f2->f2: max([0, Arg_1]) {O(n)}
4.03/2.08	
4.03/2.08	
4.03/2.08	
4.03/2.08	Sizebounds:
4.03/2.08	
4.03/2.08	`Lower:
4.03/2.08	
4.03/2.08	  0: f1->f2, Arg_0: 1 {O(1)}
4.03/2.08	
4.03/2.08	  0: f1->f2, Arg_1: 1 {O(1)}
4.03/2.08	
4.03/2.08	  1: f2->f2, Arg_0: 1 {O(1)}
4.03/2.08	
4.03/2.08	  1: f2->f2, Arg_1: 1 {O(1)}
4.03/2.08	
4.03/2.08	  2: f2->f2, Arg_0: 1 {O(1)}
4.03/2.08	
4.03/2.08	  2: f2->f2, Arg_1: 1 {O(1)}
4.03/2.08	
4.03/2.08	`Upper:
4.03/2.08	
4.03/2.08	  0: f1->f2, Arg_0: Arg_0 {O(n)}
4.03/2.08	
4.03/2.08	  0: f1->f2, Arg_1: Arg_1 {O(n)}
4.03/2.08	
4.03/2.08	  1: f2->f2, Arg_0: Arg_0 {O(n)}
4.03/2.08	
4.03/2.08	  1: f2->f2, Arg_1: Arg_1 {O(n)}
4.03/2.08	
4.03/2.08	  2: f2->f2, Arg_0: Arg_0 {O(n)}
4.03/2.08	
4.03/2.08	  2: f2->f2, Arg_1: Arg_1 {O(n)}
4.03/2.08	
4.03/2.08	
4.03/2.08	----------------------------------------
4.03/2.08	
4.03/2.08	(2)
4.03/2.08	BOUNDS(1, nat(Arg_1) + max(1, 1 + Arg_0))
4.03/2.08	
4.03/2.08	----------------------------------------
4.03/2.08	
4.03/2.08	(3) Loat Proof (FINISHED)
4.03/2.08	
4.03/2.08	
4.03/2.08	### Pre-processing the ITS problem ###
4.03/2.08	
4.03/2.08	
4.03/2.08	
4.03/2.08	Initial linear ITS problem
4.03/2.08	
4.03/2.08	   Start location: f1
4.03/2.08	
4.03/2.08	      0: f1 -> f2 : [ A>=1 && B>=1 ], cost: 1
4.03/2.08	
4.03/2.08	      1: f2 -> f2 : A'=-1+A, [ A>=2 && B>=1 ], cost: 1
4.03/2.08	
4.03/2.08	      2: f2 -> f2 : B'=-1+B, [ A>=1 && B>=2 ], cost: 1
4.03/2.08	
4.03/2.08	
4.03/2.08	
4.03/2.08	### Simplification by acceleration and chaining ###
4.03/2.08	
4.03/2.08	
4.03/2.08	
4.03/2.08	Accelerating simple loops of location 1.
4.03/2.08	
4.03/2.08	   Accelerating the following rules:
4.03/2.08	
4.03/2.08	      1: f2 -> f2 : A'=-1+A, [ A>=2 && B>=1 ], cost: 1
4.03/2.08	
4.03/2.08	      2: f2 -> f2 : B'=-1+B, [ A>=1 && B>=2 ], cost: 1
4.03/2.08	
4.03/2.08	
4.03/2.08	
4.03/2.08	   Accelerated rule 1 with metering function -1+A, yielding the new rule 3.
4.03/2.08	
4.03/2.08	   Accelerated rule 2 with metering function -1+B, yielding the new rule 4.
4.03/2.08	
4.03/2.08	   Removing the simple loops: 1 2.
4.03/2.08	
4.03/2.08	
4.03/2.08	
4.03/2.08	Accelerated all simple loops using metering functions (where possible):
4.03/2.08	
4.03/2.08	   Start location: f1
4.03/2.08	
4.03/2.08	      0: f1 -> f2 : [ A>=1 && B>=1 ], cost: 1
4.03/2.08	
4.03/2.08	      3: f2 -> f2 : A'=1, [ A>=2 && B>=1 ], cost: -1+A
4.03/2.09	
4.03/2.09	      4: f2 -> f2 : B'=1, [ A>=1 && B>=2 ], cost: -1+B
4.03/2.09	
4.03/2.09	
4.03/2.09	
4.03/2.09	Chained accelerated rules (with incoming rules):
4.03/2.09	
4.03/2.09	   Start location: f1
4.03/2.09	
4.03/2.09	      0: f1 -> f2 : [ A>=1 && B>=1 ], cost: 1
4.03/2.09	
4.03/2.09	      5: f1 -> f2 : A'=1, [ B>=1 && A>=2 ], cost: A
4.03/2.09	
4.03/2.09	      6: f1 -> f2 : B'=1, [ A>=1 && B>=2 ], cost: B
4.03/2.09	
4.03/2.09	
4.03/2.09	
4.03/2.09	Removed unreachable locations (and leaf rules with constant cost):
4.03/2.09	
4.03/2.09	   Start location: f1
4.03/2.09	
4.03/2.09	      5: f1 -> f2 : A'=1, [ B>=1 && A>=2 ], cost: A
4.03/2.09	
4.03/2.09	      6: f1 -> f2 : B'=1, [ A>=1 && B>=2 ], cost: B
4.03/2.09	
4.03/2.09	
4.03/2.09	
4.03/2.09	### Computing asymptotic complexity ###
4.03/2.09	
4.03/2.09	
4.03/2.09	
4.03/2.09	Fully simplified ITS problem
4.03/2.09	
4.03/2.09	   Start location: f1
4.03/2.09	
4.03/2.09	      5: f1 -> f2 : A'=1, [ B>=1 && A>=2 ], cost: A
4.03/2.09	
4.03/2.09	      6: f1 -> f2 : B'=1, [ A>=1 && B>=2 ], cost: B
4.03/2.09	
4.03/2.09	
4.03/2.09	
4.03/2.09	Computing asymptotic complexity for rule 5
4.03/2.09	
4.03/2.09	   Solved the limit problem by the following transformations:
4.03/2.09	
4.03/2.09	      Created initial limit problem:
4.03/2.09	
4.03/2.09	      -1+A (+/+!), A (+), B (+/+!) [not solved]
4.03/2.09	
4.03/2.09	
4.03/2.09	
4.03/2.09	      removing all constraints (solved by SMT)
4.03/2.09	
4.03/2.09	      resulting limit problem: [solved]
4.03/2.09	
4.03/2.09	
4.03/2.09	
4.03/2.09	      applying transformation rule (C) using substitution {A==n,B==n}
4.03/2.09	
4.03/2.09	      resulting limit problem:
4.03/2.09	
4.03/2.09	      [solved]
4.03/2.09	
4.03/2.09	
4.03/2.09	
4.03/2.09	   Solution:
4.03/2.09	
4.03/2.09	      A / n
4.03/2.09	
4.03/2.09	      B / n
4.03/2.09	
4.03/2.09	   Resulting cost n has complexity: Poly(n^1)
4.03/2.09	
4.03/2.09	
4.03/2.09	
4.03/2.09	Found new complexity Poly(n^1).
4.03/2.09	
4.03/2.09	
4.03/2.09	
4.03/2.09	Obtained the following overall complexity (w.r.t. the length of the input n):
4.03/2.09	
4.03/2.09	   Complexity:  Poly(n^1)
4.03/2.09	
4.03/2.09	   Cpx degree:  1
4.03/2.09	
4.03/2.09	   Solved cost: n
4.03/2.09	
4.03/2.09	   Rule cost:   A
4.03/2.09	
4.03/2.09	   Rule guard:  [ B>=1 && A>=2 ]
4.03/2.09	
4.03/2.09	
4.03/2.09	
4.03/2.09	WORST_CASE(Omega(n^1),?)
4.03/2.09	
4.03/2.09	
4.03/2.09	----------------------------------------
4.03/2.09	
4.03/2.09	(4)
4.03/2.09	BOUNDS(n^1, INF)
4.03/2.11	EOF