5.00/2.44	WORST_CASE(Omega(n^1), O(n^2))
5.00/2.45	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
5.00/2.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.00/2.45	
5.00/2.45	
5.00/2.45	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, nat(-99 * Arg_1) + max(2, 2 + -1 * Arg_1) + nat(Arg_1 * Arg_0 + -99 * Arg_1) + nat(-1 * Arg_0)).
5.00/2.45	
5.00/2.45	(0) CpxIntTrs
5.00/2.45	(1) Koat2 Proof [FINISHED, 534 ms]
5.00/2.45	(2) BOUNDS(1, nat(-99 * Arg_1) + max(2, 2 + -1 * Arg_1) + nat(Arg_1 * Arg_0 + -99 * Arg_1) + nat(-1 * Arg_0))
5.00/2.45	(3) Loat Proof [FINISHED, 739 ms]
5.00/2.45	(4) BOUNDS(n^1, INF)
5.00/2.45	
5.00/2.45	
5.00/2.45	----------------------------------------
5.00/2.45	
5.00/2.45	(0)
5.00/2.45	Obligation:
5.00/2.45	Complexity Int TRS consisting of the following rules:
5.00/2.45	f300(A, B, C) -> Com_1(f300(-(99) + A, 0, C)) :|: 0 >= A + 1 && B + 1 >= 0 && B + 1 <= 0
5.00/2.45	f300(A, B, C) -> Com_1(f300(1 + A, 1 + B, C)) :|: 0 >= A + 1 && B >= 0
5.00/2.45	f300(A, B, C) -> Com_1(f300(1 + A, 1 + B, C)) :|: 0 >= A + 1 && 0 >= 2 + B
5.00/2.45	f300(A, B, C) -> Com_1(f1(A, B, D)) :|: A >= 0
5.00/2.45	f2(A, B, C) -> Com_1(f300(A, B, C)) :|: TRUE
5.00/2.45	
5.00/2.45	The start-symbols are:[f2_3]
5.00/2.45	
5.00/2.45	
5.00/2.45	----------------------------------------
5.00/2.45	
5.00/2.45	(1) Koat2 Proof (FINISHED)
5.00/2.45	YES( ?, max([0, -99*Arg_1])+max([2, 2-Arg_1])+max([0, -(Arg_1)*(99-Arg_0)])+max([0, -(Arg_0)]) {O(n^2)})
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Initial Complexity Problem:
5.00/2.45	
5.00/2.45	  Start:  f2  
5.00/2.45	
5.00/2.45	  Program_Vars:  Arg_0, Arg_1, Arg_2  
5.00/2.45	
5.00/2.45	  Temp_Vars:  D  
5.00/2.45	
5.00/2.45	  Locations:  f1, f2, f300  
5.00/2.45	
5.00/2.45	  Transitions:
5.00/2.45	
5.00/2.45	  f2(Arg_0,Arg_1,Arg_2) -> f300(Arg_0,Arg_1,Arg_2):|:
5.00/2.45	
5.00/2.45	  f300(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1,D):|:0 <= Arg_0
5.00/2.45	
5.00/2.45	  f300(Arg_0,Arg_1,Arg_2) -> f300(-99+Arg_0,0,Arg_2):|:Arg_0+1 <= 0 && Arg_1+1 <= 0 && 0 <= 1+Arg_1
5.00/2.45	
5.00/2.45	  f300(Arg_0,Arg_1,Arg_2) -> f300(1+Arg_0,1+Arg_1,Arg_2):|:Arg_0+1 <= 0 && 0 <= Arg_1
5.00/2.45	
5.00/2.45	  f300(Arg_0,Arg_1,Arg_2) -> f300(1+Arg_0,1+Arg_1,Arg_2):|:Arg_0+1 <= 0 && 2+Arg_1 <= 0  
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Timebounds: 
5.00/2.45	
5.00/2.45	  Overall timebound: max([0, -99*Arg_1])+max([2, 2-Arg_1])+max([0, -(Arg_1)*(99-Arg_0)])+max([0, -(Arg_0)]) {O(n^2)}
5.00/2.45	
5.00/2.45	  4: f2->f300: 1 {O(1)}
5.00/2.45	
5.00/2.45	  0: f300->f300: max([0, -(Arg_1)]) {O(n)}
5.00/2.45	
5.00/2.45	  1: f300->f300: max([0, -(Arg_1)*(99-Arg_0)])+max([0, -(Arg_0)]) {O(n^2)}
5.00/2.45	
5.00/2.45	  2: f300->f300: max([0, -99*Arg_1]) {O(n)}
5.00/2.45	
5.00/2.45	  3: f300->f1: 1 {O(1)}
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Costbounds:
5.00/2.45	
5.00/2.45	  Overall costbound: max([0, -99*Arg_1])+max([2, 2-Arg_1])+max([0, -(Arg_1)*(99-Arg_0)])+max([0, -(Arg_0)]) {O(n^2)}
5.00/2.45	
5.00/2.45	  4: f2->f300: 1 {O(1)}
5.00/2.45	
5.00/2.45	  0: f300->f300: max([0, -(Arg_1)]) {O(n)}
5.00/2.45	
5.00/2.45	  1: f300->f300: max([0, -(Arg_1)*(99-Arg_0)])+max([0, -(Arg_0)]) {O(n^2)}
5.00/2.45	
5.00/2.45	  2: f300->f300: max([0, -99*Arg_1]) {O(n)}
5.00/2.45	
5.00/2.45	  3: f300->f1: 1 {O(1)}
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Sizebounds:
5.00/2.45	
5.00/2.45	`Lower:
5.00/2.45	
5.00/2.45	  4: f2->f300, Arg_0: Arg_0 {O(n)}
5.00/2.45	
5.00/2.45	  4: f2->f300, Arg_1: Arg_1 {O(n)}
5.00/2.45	
5.00/2.45	  4: f2->f300, Arg_2: Arg_2 {O(n)}
5.00/2.45	
5.00/2.45	  0: f300->f300, Arg_0: -99+Arg_0 {O(n)}
5.00/2.45	
5.00/2.45	  0: f300->f300, Arg_1: 0 {O(1)}
5.00/2.45	
5.00/2.45	  0: f300->f300, Arg_2: Arg_2 {O(n)}
5.00/2.45	
5.00/2.45	  1: f300->f300, Arg_0: min([Arg_0, -(99-Arg_0)]) {O(n)}
5.00/2.45	
5.00/2.45	  1: f300->f300, Arg_1: 1 {O(1)}
5.00/2.45	
5.00/2.45	  1: f300->f300, Arg_2: Arg_2 {O(n)}
5.00/2.45	
5.00/2.45	  2: f300->f300, Arg_0: Arg_0 {O(n)}
5.00/2.45	
5.00/2.45	  2: f300->f300, Arg_1: Arg_1 {O(n)}
5.00/2.45	
5.00/2.45	  2: f300->f300, Arg_2: Arg_2 {O(n)}
5.00/2.45	
5.00/2.45	  3: f300->f1, Arg_0: 0 {O(1)}
5.00/2.45	
5.00/2.45	  3: f300->f1, Arg_1: min([1, Arg_1]) {O(n)}
5.00/2.45	
5.00/2.45	`Upper:
5.00/2.45	
5.00/2.45	  4: f2->f300, Arg_0: Arg_0 {O(n)}
5.00/2.45	
5.00/2.45	  4: f2->f300, Arg_1: Arg_1 {O(n)}
5.00/2.45	
5.00/2.45	  4: f2->f300, Arg_2: Arg_2 {O(n)}
5.00/2.45	
5.00/2.45	  0: f300->f300, Arg_0: -100 {O(1)}
5.00/2.45	
5.00/2.45	  0: f300->f300, Arg_1: 0 {O(1)}
5.00/2.45	
5.00/2.45	  0: f300->f300, Arg_2: Arg_2 {O(n)}
5.00/2.45	
5.00/2.45	  1: f300->f300, Arg_0: 0 {O(1)}
5.00/2.45	
5.00/2.45	  1: f300->f300, Arg_1: max([0, -(Arg_1)*(99-Arg_0)])+max([0, -(Arg_0)])+max([0, Arg_1]) {O(n^2)}
5.00/2.45	
5.00/2.45	  1: f300->f300, Arg_2: Arg_2 {O(n)}
5.00/2.45	
5.00/2.45	  2: f300->f300, Arg_0: 0 {O(1)}
5.00/2.45	
5.00/2.45	  2: f300->f300, Arg_1: -1 {O(1)}
5.00/2.45	
5.00/2.45	  2: f300->f300, Arg_2: Arg_2 {O(n)}
5.00/2.45	
5.00/2.45	  3: f300->f1, Arg_0: max([0, Arg_0]) {O(n)}
5.00/2.45	
5.00/2.45	  3: f300->f1, Arg_1: max([-1, max([Arg_1, max([0, -(Arg_1)*(99-Arg_0)])+max([0, -(Arg_0)])+max([0, Arg_1])])]) {O(n^2)}
5.00/2.45	
5.00/2.45	
5.00/2.45	----------------------------------------
5.00/2.45	
5.00/2.45	(2)
5.00/2.45	BOUNDS(1, nat(-99 * Arg_1) + max(2, 2 + -1 * Arg_1) + nat(Arg_1 * Arg_0 + -99 * Arg_1) + nat(-1 * Arg_0))
5.00/2.45	
5.00/2.45	----------------------------------------
5.00/2.45	
5.00/2.45	(3) Loat Proof (FINISHED)
5.00/2.45	
5.00/2.45	
5.00/2.45	### Pre-processing the ITS problem ###
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Initial linear ITS problem
5.00/2.45	
5.00/2.45	   Start location: f2
5.00/2.45	
5.00/2.45	      0: f300 -> f300 : A'=-99+A, B'=0, [ 0>=1+A && 1+B==0 ], cost: 1
5.00/2.45	
5.00/2.45	      1: f300 -> f300 : A'=1+A, B'=1+B, [ 0>=1+A && B>=0 ], cost: 1
5.00/2.45	
5.00/2.45	      2: f300 -> f300 : A'=1+A, B'=1+B, [ 0>=1+A && 0>=2+B ], cost: 1
5.00/2.45	
5.00/2.45	      3: f300 -> f1 : C'=free, [ A>=0 ], cost: 1
5.00/2.45	
5.00/2.45	      4: f2 -> f300 : [], cost: 1
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Removed unreachable and leaf rules:
5.00/2.45	
5.00/2.45	   Start location: f2
5.00/2.45	
5.00/2.45	      0: f300 -> f300 : A'=-99+A, B'=0, [ 0>=1+A && 1+B==0 ], cost: 1
5.00/2.45	
5.00/2.45	      1: f300 -> f300 : A'=1+A, B'=1+B, [ 0>=1+A && B>=0 ], cost: 1
5.00/2.45	
5.00/2.45	      2: f300 -> f300 : A'=1+A, B'=1+B, [ 0>=1+A && 0>=2+B ], cost: 1
5.00/2.45	
5.00/2.45	      4: f2 -> f300 : [], cost: 1
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	### Simplification by acceleration and chaining ###
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Accelerating simple loops of location 0.
5.00/2.45	
5.00/2.45	   Accelerating the following rules:
5.00/2.45	
5.00/2.45	      0: f300 -> f300 : A'=-99+A, B'=0, [ 0>=1+A && 1+B==0 ], cost: 1
5.00/2.45	
5.00/2.45	      1: f300 -> f300 : A'=1+A, B'=1+B, [ 0>=1+A && B>=0 ], cost: 1
5.00/2.45	
5.00/2.45	      2: f300 -> f300 : A'=1+A, B'=1+B, [ 0>=1+A && 0>=2+B ], cost: 1
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	   Accelerated rule 0 with metering function -B, yielding the new rule 5.
5.00/2.45	
5.00/2.45	   Accelerated rule 1 with metering function -A, yielding the new rule 6.
5.00/2.45	
5.00/2.45	   Accelerated rule 2 with metering function -1-A (after adding A>=B), yielding the new rule 7.
5.00/2.45	
5.00/2.45	   Accelerated rule 2 with metering function -1-B (after adding A<=B), yielding the new rule 8.
5.00/2.45	
5.00/2.45	   Removing the simple loops: 0 1 2.
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Accelerated all simple loops using metering functions (where possible):
5.00/2.45	
5.00/2.45	   Start location: f2
5.00/2.45	
5.00/2.45	      5: f300 -> f300 : A'=A+99*B, B'=0, [ 0>=1+A && 1+B==0 ], cost: -B
5.00/2.45	
5.00/2.45	      6: f300 -> f300 : A'=0, B'=-A+B, [ 0>=1+A && B>=0 ], cost: -A
5.00/2.45	
5.00/2.45	      7: f300 -> f300 : A'=-1, B'=-1-A+B, [ 0>=2+B && A>=B && -1-A>=1 ], cost: -1-A
5.00/2.45	
5.00/2.45	      8: f300 -> f300 : A'=-1+A-B, B'=-1, [ 0>=1+A && 0>=2+B && A<=B ], cost: -1-B
5.00/2.45	
5.00/2.45	      4: f2 -> f300 : [], cost: 1
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Chained accelerated rules (with incoming rules):
5.00/2.45	
5.00/2.45	   Start location: f2
5.00/2.45	
5.00/2.45	      4: f2 -> f300 : [], cost: 1
5.00/2.45	
5.00/2.45	      9: f2 -> f300 : A'=A+99*B, B'=0, [ 0>=1+A && 1+B==0 ], cost: 1-B
5.00/2.45	
5.00/2.45	     10: f2 -> f300 : A'=0, B'=-A+B, [ 0>=1+A && B>=0 ], cost: 1-A
5.00/2.45	
5.00/2.45	     11: f2 -> f300 : A'=-1, B'=-1-A+B, [ 0>=2+B && A>=B && -1-A>=1 ], cost: -A
5.00/2.45	
5.00/2.45	     12: f2 -> f300 : A'=-1+A-B, B'=-1, [ 0>=1+A && 0>=2+B && A<=B ], cost: -B
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Removed unreachable locations (and leaf rules with constant cost):
5.00/2.45	
5.00/2.45	   Start location: f2
5.00/2.45	
5.00/2.45	      9: f2 -> f300 : A'=A+99*B, B'=0, [ 0>=1+A && 1+B==0 ], cost: 1-B
5.00/2.45	
5.00/2.45	     10: f2 -> f300 : A'=0, B'=-A+B, [ 0>=1+A && B>=0 ], cost: 1-A
5.00/2.45	
5.00/2.45	     11: f2 -> f300 : A'=-1, B'=-1-A+B, [ 0>=2+B && A>=B && -1-A>=1 ], cost: -A
5.00/2.45	
5.00/2.45	     12: f2 -> f300 : A'=-1+A-B, B'=-1, [ 0>=1+A && 0>=2+B && A<=B ], cost: -B
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	### Computing asymptotic complexity ###
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Fully simplified ITS problem
5.00/2.45	
5.00/2.45	   Start location: f2
5.00/2.45	
5.00/2.45	      9: f2 -> f300 : A'=A+99*B, B'=0, [ 0>=1+A && 1+B==0 ], cost: 1-B
5.00/2.45	
5.00/2.45	     10: f2 -> f300 : A'=0, B'=-A+B, [ 0>=1+A && B>=0 ], cost: 1-A
5.00/2.45	
5.00/2.45	     11: f2 -> f300 : A'=-1, B'=-1-A+B, [ 0>=2+B && A>=B && -1-A>=1 ], cost: -A
5.00/2.45	
5.00/2.45	     12: f2 -> f300 : A'=-1+A-B, B'=-1, [ 0>=1+A && 0>=2+B && A<=B ], cost: -B
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Computing asymptotic complexity for rule 9
5.00/2.45	
5.00/2.45	   Could not solve the limit problem.
5.00/2.45	
5.00/2.45	   Resulting cost 0 has complexity: Unknown
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Computing asymptotic complexity for rule 10
5.00/2.45	
5.00/2.45	   Solved the limit problem by the following transformations:
5.00/2.45	
5.00/2.45	      Created initial limit problem:
5.00/2.45	
5.00/2.45	      1-A (+), 1+B (+/+!), -A (+/+!) [not solved]
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	      applying transformation rule (C) using substitution {B==0}
5.00/2.45	
5.00/2.45	      resulting limit problem:
5.00/2.45	
5.00/2.45	      1 (+/+!), 1-A (+), -A (+/+!) [not solved]
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	      applying transformation rule (B), deleting 1 (+/+!)
5.00/2.45	
5.00/2.45	      resulting limit problem:
5.00/2.45	
5.00/2.45	      1-A (+), -A (+/+!) [not solved]
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	      removing all constraints (solved by SMT)
5.00/2.45	
5.00/2.45	      resulting limit problem: [solved]
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	      applying transformation rule (C) using substitution {A==-n}
5.00/2.45	
5.00/2.45	      resulting limit problem:
5.00/2.45	
5.00/2.45	      [solved]
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	   Solution:
5.00/2.45	
5.00/2.45	      A / -n
5.00/2.45	
5.00/2.45	      B / 0
5.00/2.45	
5.00/2.45	   Resulting cost 1+n has complexity: Poly(n^1)
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Found new complexity Poly(n^1).
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	Obtained the following overall complexity (w.r.t. the length of the input n):
5.00/2.45	
5.00/2.45	   Complexity:  Poly(n^1)
5.00/2.45	
5.00/2.45	   Cpx degree:  1
5.00/2.45	
5.00/2.45	   Solved cost: 1+n
5.00/2.45	
5.00/2.45	   Rule cost:   1-A
5.00/2.45	
5.00/2.45	   Rule guard:  [ 0>=1+A && B>=0 ]
5.00/2.45	
5.00/2.45	
5.00/2.45	
5.00/2.45	WORST_CASE(Omega(n^1),?)
5.00/2.45	
5.00/2.45	
5.00/2.45	----------------------------------------
5.00/2.45	
5.00/2.45	(4)
5.00/2.45	BOUNDS(n^1, INF)
5.09/2.49	EOF