4.13/1.98	WORST_CASE(Omega(n^1), O(n^2))
4.13/1.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.13/1.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.13/1.98	
4.13/1.98	
4.13/1.98	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_1 + -1 * Arg_2) + nat(Arg_0 + -1 * Arg_1) + nat(Arg_0 * Arg_1 + -1 * Arg_0 * Arg_2 + Arg_0 * nat(Arg_0 + -1 * Arg_1) + Arg_1 * Arg_2 + -1 * Arg_1 * nat(Arg_0 + -1 * Arg_1) + -1 * Arg_1^2)).
4.13/1.98	
4.13/1.98	(0) CpxIntTrs
4.13/1.98	(1) Koat2 Proof [FINISHED, 201 ms]
4.13/1.98	(2) BOUNDS(1, max(1, 1 + Arg_1 + -1 * Arg_2) + nat(Arg_0 + -1 * Arg_1) + nat(Arg_0 * Arg_1 + -1 * Arg_0 * Arg_2 + Arg_0 * nat(Arg_0 + -1 * Arg_1) + Arg_1 * Arg_2 + -1 * Arg_1 * nat(Arg_0 + -1 * Arg_1) + -1 * Arg_1^2))
4.13/1.98	(3) Loat Proof [FINISHED, 314 ms]
4.13/1.98	(4) BOUNDS(n^1, INF)
4.13/1.98	
4.13/1.98	
4.13/1.98	----------------------------------------
4.13/1.98	
4.13/1.98	(0)
4.13/1.98	Obligation:
4.13/1.98	Complexity Int TRS consisting of the following rules:
4.13/1.98	f1(A, B, C) -> Com_1(f1(A, B + 1, C)) :|: A >= B + 1
4.13/1.98	f3(A, B, C) -> Com_1(f1(A, B, C)) :|: B >= C + 1
4.13/1.98	f1(A, B, C) -> Com_1(f1(A, B, C + 1)) :|: B >= C + 2 && B >= A
4.13/1.99	
4.13/1.99	The start-symbols are:[f3_3]
4.13/1.99	
4.13/1.99	
4.13/1.99	----------------------------------------
4.13/1.99	
4.13/1.99	(1) Koat2 Proof (FINISHED)
4.13/1.99	YES( ?, 1+max([0, Arg_1-Arg_2])+max([0, (Arg_0-Arg_1)*(Arg_1-Arg_2+max([0, Arg_0-Arg_1]))])+max([0, Arg_0-Arg_1]) {O(n^2)})
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	Initial Complexity Problem:
4.13/1.99	
4.13/1.99	  Start:  f3  
4.13/1.99	
4.13/1.99	  Program_Vars:  Arg_0, Arg_1, Arg_2  
4.13/1.99	
4.13/1.99	  Temp_Vars:    
4.13/1.99	
4.13/1.99	  Locations:  f1, f3  
4.13/1.99	
4.13/1.99	  Transitions:
4.13/1.99	
4.13/1.99	  f1(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1+1,Arg_2):|:1+Arg_2 <= Arg_1 && Arg_1+1 <= Arg_0
4.13/1.99	
4.13/1.99	  f1(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1,Arg_2+1):|:1+Arg_2 <= Arg_1 && Arg_2+2 <= Arg_1 && Arg_0 <= Arg_1
4.13/1.99	
4.13/1.99	  f3(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1,Arg_2):|:Arg_2+1 <= Arg_1  
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	Timebounds: 
4.13/1.99	
4.13/1.99	  Overall timebound: 1+max([0, Arg_1-Arg_2])+max([0, (Arg_0-Arg_1)*(Arg_1-Arg_2+max([0, Arg_0-Arg_1]))])+max([0, Arg_0-Arg_1]) {O(n^2)}
4.13/1.99	
4.13/1.99	  0: f1->f1: max([0, Arg_0-Arg_1]) {O(n)}
4.13/1.99	
4.13/1.99	  2: f1->f1: max([0, Arg_1-Arg_2])+max([0, (Arg_0-Arg_1)*(Arg_1-Arg_2+max([0, Arg_0-Arg_1]))]) {O(n^2)}
4.13/1.99	
4.13/1.99	  1: f3->f1: 1 {O(1)}
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	Costbounds:
4.13/1.99	
4.13/1.99	  Overall costbound: 1+max([0, Arg_1-Arg_2])+max([0, (Arg_0-Arg_1)*(Arg_1-Arg_2+max([0, Arg_0-Arg_1]))])+max([0, Arg_0-Arg_1]) {O(n^2)}
4.13/1.99	
4.13/1.99	  0: f1->f1: max([0, Arg_0-Arg_1]) {O(n)}
4.13/1.99	
4.13/1.99	  2: f1->f1: max([0, Arg_1-Arg_2])+max([0, (Arg_0-Arg_1)*(Arg_1-Arg_2+max([0, Arg_0-Arg_1]))]) {O(n^2)}
4.13/1.99	
4.13/1.99	  1: f3->f1: 1 {O(1)}
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	Sizebounds:
4.13/1.99	
4.13/1.99	`Lower:
4.13/1.99	
4.13/1.99	  0: f1->f1, Arg_0: Arg_0 {O(n)}
4.13/1.99	
4.13/1.99	  0: f1->f1, Arg_1: Arg_1 {O(n)}
4.13/1.99	
4.13/1.99	  0: f1->f1, Arg_2: Arg_2 {O(n)}
4.13/1.99	
4.13/1.99	  2: f1->f1, Arg_0: Arg_0 {O(n)}
4.13/1.99	
4.13/1.99	  2: f1->f1, Arg_1: Arg_1 {O(n)}
4.13/1.99	
4.13/1.99	  2: f1->f1, Arg_2: Arg_2 {O(n)}
4.13/1.99	
4.13/1.99	  1: f3->f1, Arg_0: Arg_0 {O(n)}
4.13/1.99	
4.13/1.99	  1: f3->f1, Arg_1: Arg_1 {O(n)}
4.13/1.99	
4.13/1.99	  1: f3->f1, Arg_2: Arg_2 {O(n)}
4.13/1.99	
4.13/1.99	`Upper:
4.13/1.99	
4.13/1.99	  0: f1->f1, Arg_0: Arg_0 {O(n)}
4.13/1.99	
4.13/1.99	  0: f1->f1, Arg_1: Arg_1+max([0, Arg_0-Arg_1]) {O(n)}
4.13/1.99	
4.13/1.99	  0: f1->f1, Arg_2: Arg_2 {O(n)}
4.13/1.99	
4.13/1.99	  2: f1->f1, Arg_0: Arg_0 {O(n)}
4.13/1.99	
4.13/1.99	  2: f1->f1, Arg_1: max([Arg_1, Arg_1+max([0, Arg_0-Arg_1])]) {O(n)}
4.13/1.99	
4.13/1.99	  2: f1->f1, Arg_2: Arg_2+max([0, Arg_1-Arg_2])+max([0, (Arg_0-Arg_1)*(Arg_1-Arg_2+max([0, Arg_0-Arg_1]))]) {O(n^2)}
4.13/1.99	
4.13/1.99	  1: f3->f1, Arg_0: Arg_0 {O(n)}
4.13/1.99	
4.13/1.99	  1: f3->f1, Arg_1: Arg_1 {O(n)}
4.13/1.99	
4.13/1.99	  1: f3->f1, Arg_2: Arg_2 {O(n)}
4.13/1.99	
4.13/1.99	
4.13/1.99	----------------------------------------
4.13/1.99	
4.13/1.99	(2)
4.13/1.99	BOUNDS(1, max(1, 1 + Arg_1 + -1 * Arg_2) + nat(Arg_0 + -1 * Arg_1) + nat(Arg_0 * Arg_1 + -1 * Arg_0 * Arg_2 + Arg_0 * nat(Arg_0 + -1 * Arg_1) + Arg_1 * Arg_2 + -1 * Arg_1 * nat(Arg_0 + -1 * Arg_1) + -1 * Arg_1^2))
4.13/1.99	
4.13/1.99	----------------------------------------
4.13/1.99	
4.13/1.99	(3) Loat Proof (FINISHED)
4.13/1.99	
4.13/1.99	
4.13/1.99	### Pre-processing the ITS problem ###
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	Initial linear ITS problem
4.13/1.99	
4.13/1.99	   Start location: f3
4.13/1.99	
4.13/1.99	      0: f1 -> f1 : B'=1+B, [ A>=1+B ], cost: 1
4.13/1.99	
4.13/1.99	      2: f1 -> f1 : C'=1+C, [ B>=2+C && B>=A ], cost: 1
4.13/1.99	
4.13/1.99	      1: f3 -> f1 : [ B>=1+C ], cost: 1
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	### Simplification by acceleration and chaining ###
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	Accelerating simple loops of location 0.
4.13/1.99	
4.13/1.99	   Accelerating the following rules:
4.13/1.99	
4.13/1.99	      0: f1 -> f1 : B'=1+B, [ A>=1+B ], cost: 1
4.13/1.99	
4.13/1.99	      2: f1 -> f1 : C'=1+C, [ B>=2+C && B>=A ], cost: 1
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	   Accelerated rule 0 with metering function A-B, yielding the new rule 3.
4.13/1.99	
4.13/1.99	   Accelerated rule 2 with metering function -1-C+B, yielding the new rule 4.
4.13/1.99	
4.13/1.99	   Removing the simple loops: 0 2.
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	Accelerated all simple loops using metering functions (where possible):
4.13/1.99	
4.13/1.99	   Start location: f3
4.13/1.99	
4.13/1.99	      3: f1 -> f1 : B'=A, [ A>=1+B ], cost: A-B
4.13/1.99	
4.13/1.99	      4: f1 -> f1 : C'=-1+B, [ B>=2+C && B>=A ], cost: -1-C+B
4.13/1.99	
4.13/1.99	      1: f3 -> f1 : [ B>=1+C ], cost: 1
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	Chained accelerated rules (with incoming rules):
4.13/1.99	
4.13/1.99	   Start location: f3
4.13/1.99	
4.13/1.99	      1: f3 -> f1 : [ B>=1+C ], cost: 1
4.13/1.99	
4.13/1.99	      5: f3 -> f1 : B'=A, [ B>=1+C && A>=1+B ], cost: 1+A-B
4.13/1.99	
4.13/1.99	      6: f3 -> f1 : C'=-1+B, [ B>=2+C && B>=A ], cost: -C+B
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	Removed unreachable locations (and leaf rules with constant cost):
4.13/1.99	
4.13/1.99	   Start location: f3
4.13/1.99	
4.13/1.99	      5: f3 -> f1 : B'=A, [ B>=1+C && A>=1+B ], cost: 1+A-B
4.13/1.99	
4.13/1.99	      6: f3 -> f1 : C'=-1+B, [ B>=2+C && B>=A ], cost: -C+B
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	### Computing asymptotic complexity ###
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	Fully simplified ITS problem
4.13/1.99	
4.13/1.99	   Start location: f3
4.13/1.99	
4.13/1.99	      5: f3 -> f1 : B'=A, [ B>=1+C && A>=1+B ], cost: 1+A-B
4.13/1.99	
4.13/1.99	      6: f3 -> f1 : C'=-1+B, [ B>=2+C && B>=A ], cost: -C+B
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	Computing asymptotic complexity for rule 5
4.13/1.99	
4.13/1.99	   Solved the limit problem by the following transformations:
4.13/1.99	
4.13/1.99	      Created initial limit problem:
4.13/1.99	
4.13/1.99	      -C+B (+/+!), 1+A-B (+), A-B (+/+!) [not solved]
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	      applying transformation rule (C) using substitution {B==1+C}
4.13/1.99	
4.13/1.99	      resulting limit problem:
4.13/1.99	
4.13/1.99	      1 (+/+!), -1-C+A (+/+!), -C+A (+) [not solved]
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	      applying transformation rule (C) using substitution {A==1+B}
4.13/1.99	
4.13/1.99	      resulting limit problem:
4.13/1.99	
4.13/1.99	      1 (+/+!), 1-C+B (+), -C+B (+/+!) [not solved]
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	      applying transformation rule (B), deleting 1 (+/+!)
4.13/1.99	
4.13/1.99	      resulting limit problem:
4.13/1.99	
4.13/1.99	      1-C+B (+), -C+B (+/+!) [not solved]
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	      removing all constraints (solved by SMT)
4.13/1.99	
4.13/1.99	      resulting limit problem: [solved]
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	      applying transformation rule (C) using substitution {C==-2*n,B==-n}
4.13/1.99	
4.13/1.99	      resulting limit problem:
4.13/1.99	
4.13/1.99	      [solved]
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	   Solution:
4.13/1.99	
4.13/1.99	      C / -2*n
4.13/1.99	
4.13/1.99	      A / 1-n
4.13/1.99	
4.13/1.99	      B / 1-2*n
4.13/1.99	
4.13/1.99	   Resulting cost 1+n has complexity: Poly(n^1)
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	Found new complexity Poly(n^1).
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	Obtained the following overall complexity (w.r.t. the length of the input n):
4.13/1.99	
4.13/1.99	   Complexity:  Poly(n^1)
4.13/1.99	
4.13/1.99	   Cpx degree:  1
4.13/1.99	
4.13/1.99	   Solved cost: 1+n
4.13/1.99	
4.13/1.99	   Rule cost:   1+A-B
4.13/1.99	
4.13/1.99	   Rule guard:  [ B>=1+C && A>=1+B ]
4.13/1.99	
4.13/1.99	
4.13/1.99	
4.13/1.99	WORST_CASE(Omega(n^1),?)
4.13/1.99	
4.13/1.99	
4.13/1.99	----------------------------------------
4.13/1.99	
4.13/1.99	(4)
4.13/1.99	BOUNDS(n^1, INF)
4.34/2.00	EOF