4.11/2.06	WORST_CASE(Omega(n^2), O(n^2))
4.11/2.07	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.11/2.07	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.11/2.07	
4.11/2.07	
4.11/2.07	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, max(1, 1 + -1 * Arg_0 + Arg_0^2) + nat(2 * Arg_0)).
4.11/2.07	
4.11/2.07	(0) CpxIntTrs
4.11/2.07	(1) Koat2 Proof [FINISHED, 143 ms]
4.11/2.07	(2) BOUNDS(1, max(1, 1 + -1 * Arg_0 + Arg_0^2) + nat(2 * Arg_0))
4.11/2.07	(3) Loat Proof [FINISHED, 422 ms]
4.11/2.07	(4) BOUNDS(n^2, INF)
4.11/2.07	
4.11/2.07	
4.11/2.07	----------------------------------------
4.11/2.07	
4.11/2.07	(0)
4.11/2.07	Obligation:
4.11/2.07	Complexity Int TRS consisting of the following rules:
4.11/2.07	f1(A, B) -> Com_1(f3(A, A)) :|: A >= 1
4.11/2.07	f3(A, B) -> Com_1(f3(A, B - 1)) :|: B >= 1
4.11/2.07	f3(A, B) -> Com_1(f3(A - 1, A - 1)) :|: 0 >= B && A >= 2
4.11/2.07	
4.11/2.07	The start-symbols are:[f1_2]
4.11/2.07	
4.11/2.07	
4.11/2.07	----------------------------------------
4.11/2.07	
4.11/2.07	(1) Koat2 Proof (FINISHED)
4.11/2.07	YES( ?, max([1, 1+Arg_0*(-1+Arg_0)])+max([0, Arg_0])+max([0, Arg_0]) {O(n^2)})
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Initial Complexity Problem:
4.11/2.07	
4.11/2.07	  Start:  f1  
4.11/2.07	
4.11/2.07	  Program_Vars:  Arg_0, Arg_1  
4.11/2.07	
4.11/2.07	  Temp_Vars:    
4.11/2.07	
4.11/2.07	  Locations:  f1, f3  
4.11/2.07	
4.11/2.07	  Transitions:
4.11/2.07	
4.11/2.07	  f1(Arg_0,Arg_1) -> f3(Arg_0,Arg_0):|:1 <= Arg_0
4.11/2.07	
4.11/2.07	  f3(Arg_0,Arg_1) -> f3(Arg_0,Arg_1-1):|:Arg_1 <= Arg_0 && 1 <= Arg_0 && 1 <= Arg_1
4.11/2.07	
4.11/2.07	  f3(Arg_0,Arg_1) -> f3(Arg_0-1,Arg_0-1):|:Arg_1 <= Arg_0 && 1 <= Arg_0 && Arg_1 <= 0 && 2 <= Arg_0  
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Timebounds: 
4.11/2.07	
4.11/2.07	  Overall timebound: max([1, 1+Arg_0*(-1+Arg_0)])+max([0, Arg_0])+max([0, Arg_0]) {O(n^2)}
4.11/2.07	
4.11/2.07	  0: f1->f3: 1 {O(1)}
4.11/2.07	
4.11/2.07	  1: f3->f3: max([0, Arg_0*(-1+Arg_0)])+max([0, Arg_0]) {O(n^2)}
4.11/2.07	
4.11/2.07	  2: f3->f3: max([0, Arg_0]) {O(n)}
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Costbounds:
4.11/2.07	
4.11/2.07	  Overall costbound: max([1, 1+Arg_0*(-1+Arg_0)])+max([0, Arg_0])+max([0, Arg_0]) {O(n^2)}
4.11/2.07	
4.11/2.07	  0: f1->f3: 1 {O(1)}
4.11/2.07	
4.11/2.07	  1: f3->f3: max([0, Arg_0*(-1+Arg_0)])+max([0, Arg_0]) {O(n^2)}
4.11/2.07	
4.11/2.07	  2: f3->f3: max([0, Arg_0]) {O(n)}
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Sizebounds:
4.11/2.07	
4.11/2.07	`Lower:
4.11/2.07	
4.11/2.07	  0: f1->f3, Arg_0: 1 {O(1)}
4.11/2.07	
4.11/2.07	  0: f1->f3, Arg_1: 1 {O(1)}
4.11/2.07	
4.11/2.07	  1: f3->f3, Arg_0: 1 {O(1)}
4.11/2.07	
4.11/2.07	  1: f3->f3, Arg_1: 0 {O(1)}
4.11/2.07	
4.11/2.07	  2: f3->f3, Arg_0: 1 {O(1)}
4.11/2.07	
4.11/2.07	  2: f3->f3, Arg_1: 1 {O(1)}
4.11/2.07	
4.11/2.07	`Upper:
4.11/2.07	
4.11/2.07	  0: f1->f3, Arg_0: Arg_0 {O(n)}
4.11/2.07	
4.11/2.07	  0: f1->f3, Arg_1: Arg_0 {O(n)}
4.11/2.07	
4.11/2.07	  1: f3->f3, Arg_0: Arg_0 {O(n)}
4.11/2.07	
4.11/2.07	  1: f3->f3, Arg_1: max([Arg_0, -1+Arg_0]) {O(n)}
4.11/2.07	
4.11/2.07	  2: f3->f3, Arg_0: Arg_0 {O(n)}
4.11/2.07	
4.11/2.07	  2: f3->f3, Arg_1: -1+Arg_0 {O(n)}
4.11/2.07	
4.11/2.07	
4.11/2.07	----------------------------------------
4.11/2.07	
4.11/2.07	(2)
4.11/2.07	BOUNDS(1, max(1, 1 + -1 * Arg_0 + Arg_0^2) + nat(2 * Arg_0))
4.11/2.07	
4.11/2.07	----------------------------------------
4.11/2.07	
4.11/2.07	(3) Loat Proof (FINISHED)
4.11/2.07	
4.11/2.07	
4.11/2.07	### Pre-processing the ITS problem ###
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Initial linear ITS problem
4.11/2.07	
4.11/2.07	   Start location: f1
4.11/2.07	
4.11/2.07	      0: f1 -> f3 : B'=A, [ A>=1 ], cost: 1
4.11/2.07	
4.11/2.07	      1: f3 -> f3 : B'=-1+B, [ B>=1 ], cost: 1
4.11/2.07	
4.11/2.07	      2: f3 -> f3 : A'=-1+A, B'=-1+A, [ 0>=B && A>=2 ], cost: 1
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	### Simplification by acceleration and chaining ###
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Accelerating simple loops of location 1.
4.11/2.07	
4.11/2.07	   Accelerating the following rules:
4.11/2.07	
4.11/2.07	      1: f3 -> f3 : B'=-1+B, [ B>=1 ], cost: 1
4.11/2.07	
4.11/2.07	      2: f3 -> f3 : A'=-1+A, B'=-1+A, [ 0>=B && A>=2 ], cost: 1
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	   Accelerated rule 1 with metering function B, yielding the new rule 3.
4.11/2.07	
4.11/2.07	   Found no metering function for rule 2.
4.11/2.07	
4.11/2.07	   Nested simple loops 2 (outer loop) and 3 (inner loop) with metering function -1+A, resulting in the new rules: 4, 5.
4.11/2.07	
4.11/2.07	   Removing the simple loops: 1 2.
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Accelerated all simple loops using metering functions (where possible):
4.11/2.07	
4.11/2.07	   Start location: f1
4.11/2.07	
4.11/2.07	      0: f1 -> f3 : B'=A, [ A>=1 ], cost: 1
4.11/2.07	
4.11/2.07	      3: f3 -> f3 : B'=0, [ B>=1 ], cost: B
4.11/2.07	
4.11/2.07	      4: f3 -> f3 : A'=1, B'=1, [ B>=1 && A>=2 ], cost: -3/2+(-1+A)*A+3/2*A-1/2*(-1+A)^2
4.11/2.07	
4.11/2.07	      5: f3 -> f3 : A'=1, B'=1, [ 0>=B && -1+A>=2 ], cost: -2-1/2*(-2+A)^2+3/2*A+(-1+A)*(-2+A)
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Chained accelerated rules (with incoming rules):
4.11/2.07	
4.11/2.07	   Start location: f1
4.11/2.07	
4.11/2.07	      0: f1 -> f3 : B'=A, [ A>=1 ], cost: 1
4.11/2.07	
4.11/2.07	      6: f1 -> f3 : B'=0, [ A>=1 ], cost: 1+A
4.11/2.07	
4.11/2.07	      7: f1 -> f3 : A'=1, B'=1, [ A>=2 ], cost: -1/2+(-1+A)*A+3/2*A-1/2*(-1+A)^2
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Removed unreachable locations (and leaf rules with constant cost):
4.11/2.07	
4.11/2.07	   Start location: f1
4.11/2.07	
4.11/2.07	      6: f1 -> f3 : B'=0, [ A>=1 ], cost: 1+A
4.11/2.07	
4.11/2.07	      7: f1 -> f3 : A'=1, B'=1, [ A>=2 ], cost: -1/2+(-1+A)*A+3/2*A-1/2*(-1+A)^2
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	### Computing asymptotic complexity ###
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Fully simplified ITS problem
4.11/2.07	
4.11/2.07	   Start location: f1
4.11/2.07	
4.11/2.07	      6: f1 -> f3 : B'=0, [ A>=1 ], cost: 1+A
4.11/2.07	
4.11/2.07	      7: f1 -> f3 : A'=1, B'=1, [ A>=2 ], cost: -1/2+(-1+A)*A+3/2*A-1/2*(-1+A)^2
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Computing asymptotic complexity for rule 6
4.11/2.07	
4.11/2.07	   Solved the limit problem by the following transformations:
4.11/2.07	
4.11/2.07	      Created initial limit problem:
4.11/2.07	
4.11/2.07	      A (+/+!), 1+A (+) [not solved]
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	      removing all constraints (solved by SMT)
4.11/2.07	
4.11/2.07	      resulting limit problem: [solved]
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	      applying transformation rule (C) using substitution {A==n}
4.11/2.07	
4.11/2.07	      resulting limit problem:
4.11/2.07	
4.11/2.07	      [solved]
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	   Solution:
4.11/2.07	
4.11/2.07	      A / n
4.11/2.07	
4.11/2.07	   Resulting cost 1+n has complexity: Poly(n^1)
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Found new complexity Poly(n^1).
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Computing asymptotic complexity for rule 7
4.11/2.07	
4.11/2.07	   Solved the limit problem by the following transformations:
4.11/2.07	
4.11/2.07	      Created initial limit problem:
4.11/2.07	
4.11/2.07	      -1+A (+/+!), -1+3/2*A+1/2*A^2 (+) [not solved]
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	      removing all constraints (solved by SMT)
4.11/2.07	
4.11/2.07	      resulting limit problem: [solved]
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	      applying transformation rule (C) using substitution {A==n}
4.11/2.07	
4.11/2.07	      resulting limit problem:
4.11/2.07	
4.11/2.07	      [solved]
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	   Solution:
4.11/2.07	
4.11/2.07	      A / n
4.11/2.07	
4.11/2.07	   Resulting cost -1+3/2*n+1/2*n^2 has complexity: Poly(n^2)
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Found new complexity Poly(n^2).
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	Obtained the following overall complexity (w.r.t. the length of the input n):
4.11/2.07	
4.11/2.07	   Complexity:  Poly(n^2)
4.11/2.07	
4.11/2.07	   Cpx degree:  2
4.11/2.07	
4.11/2.07	   Solved cost: -1+3/2*n+1/2*n^2
4.11/2.07	
4.11/2.07	   Rule cost:   -1/2+(-1+A)*A+3/2*A-1/2*(-1+A)^2
4.11/2.07	
4.11/2.07	   Rule guard:  [ A>=2 ]
4.11/2.07	
4.11/2.07	
4.11/2.07	
4.11/2.07	WORST_CASE(Omega(n^2),?)
4.11/2.07	
4.11/2.07	
4.11/2.07	----------------------------------------
4.11/2.07	
4.11/2.07	(4)
4.11/2.07	BOUNDS(n^2, INF)
4.22/2.10	EOF