4.13/2.00	WORST_CASE(Omega(n^2), O(n^2))
4.13/2.01	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.13/2.01	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.13/2.01	
4.13/2.01	
4.13/2.01	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, max(max(Arg_1, Arg_0) + nat(Arg_0^2), Arg_1, 0) + max(2, 2 + Arg_0)).
4.13/2.01	
4.13/2.01	(0) CpxIntTrs
4.13/2.01	(1) Koat2 Proof [FINISHED, 237 ms]
4.13/2.01	(2) BOUNDS(1, max(max(Arg_1, Arg_0) + nat(Arg_0^2), Arg_1, 0) + max(2, 2 + Arg_0))
4.13/2.01	(3) Loat Proof [FINISHED, 333 ms]
4.13/2.01	(4) BOUNDS(n^2, INF)
4.13/2.01	
4.13/2.01	
4.13/2.01	----------------------------------------
4.13/2.01	
4.13/2.01	(0)
4.13/2.01	Obligation:
4.13/2.01	Complexity Int TRS consisting of the following rules:
4.13/2.01	l0(A, B) -> Com_1(l1(A, B)) :|: TRUE
4.13/2.01	l1(A, B) -> Com_1(l1(A - 1, B + A)) :|: A >= 1
4.13/2.01	l1(A, B) -> Com_1(l2(A, B)) :|: 0 >= A
4.13/2.01	l2(A, B) -> Com_1(l2(A, B - 1)) :|: B >= 1
4.13/2.01	
4.13/2.01	The start-symbols are:[l0_2]
4.13/2.01	
4.13/2.01	
4.13/2.01	----------------------------------------
4.13/2.01	
4.13/2.01	(1) Koat2 Proof (FINISHED)
4.13/2.01	YES( ?, max([0, max([Arg_1, max([Arg_0, max([Arg_0, Arg_1])])+max([0, Arg_0*Arg_0])])])+max([2, 2+Arg_0]) {O(n^2)})
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Initial Complexity Problem:
4.13/2.01	
4.13/2.01	  Start:  l0  
4.13/2.01	
4.13/2.01	  Program_Vars:  Arg_0, Arg_1  
4.13/2.01	
4.13/2.01	  Temp_Vars:    
4.13/2.01	
4.13/2.01	  Locations:  l0, l1, l2  
4.13/2.01	
4.13/2.01	  Transitions:
4.13/2.01	
4.13/2.01	  l0(Arg_0,Arg_1) -> l1(Arg_0,Arg_1):|:
4.13/2.01	
4.13/2.01	  l1(Arg_0,Arg_1) -> l1(Arg_0-1,Arg_1+Arg_0):|:1 <= Arg_0
4.13/2.01	
4.13/2.01	  l1(Arg_0,Arg_1) -> l2(Arg_0,Arg_1):|:Arg_0 <= 0
4.13/2.01	
4.13/2.01	  l2(Arg_0,Arg_1) -> l2(Arg_0,Arg_1-1):|:Arg_0 <= 0 && 1 <= Arg_1  
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Timebounds: 
4.13/2.01	
4.13/2.01	  Overall timebound: max([0, max([Arg_1, max([Arg_0, max([Arg_0, Arg_1])])+max([0, Arg_0*Arg_0])])])+max([2, 2+Arg_0]) {O(n^2)}
4.13/2.01	
4.13/2.01	  0: l0->l1: 1 {O(1)}
4.13/2.01	
4.13/2.01	  1: l1->l1: max([0, Arg_0]) {O(n)}
4.13/2.01	
4.13/2.01	  2: l1->l2: 1 {O(1)}
4.13/2.01	
4.13/2.01	  3: l2->l2: max([0, max([Arg_1, max([Arg_0, max([Arg_0, Arg_1])])+max([0, Arg_0*Arg_0])])]) {O(n^2)}
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Costbounds:
4.13/2.01	
4.13/2.01	  Overall costbound: max([0, max([Arg_1, max([Arg_0, max([Arg_0, Arg_1])])+max([0, Arg_0*Arg_0])])])+max([2, 2+Arg_0]) {O(n^2)}
4.13/2.01	
4.13/2.01	  0: l0->l1: 1 {O(1)}
4.13/2.01	
4.13/2.01	  1: l1->l1: max([0, Arg_0]) {O(n)}
4.13/2.01	
4.13/2.01	  2: l1->l2: 1 {O(1)}
4.13/2.01	
4.13/2.01	  3: l2->l2: max([0, max([Arg_1, max([Arg_0, max([Arg_0, Arg_1])])+max([0, Arg_0*Arg_0])])]) {O(n^2)}
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Sizebounds:
4.13/2.01	
4.13/2.01	`Lower:
4.13/2.01	
4.13/2.01	  0: l0->l1, Arg_0: Arg_0 {O(n)}
4.13/2.01	
4.13/2.01	  0: l0->l1, Arg_1: Arg_1 {O(n)}
4.13/2.01	
4.13/2.01	  1: l1->l1, Arg_0: 0 {O(1)}
4.13/2.01	
4.13/2.01	  1: l1->l1, Arg_1: Arg_1 {O(n)}
4.13/2.01	
4.13/2.01	  2: l1->l2, Arg_0: min([0, Arg_0]) {O(n)}
4.13/2.01	
4.13/2.01	  2: l1->l2, Arg_1: Arg_1 {O(n)}
4.13/2.01	
4.13/2.01	  3: l2->l2, Arg_0: min([0, Arg_0]) {O(n)}
4.13/2.01	
4.13/2.01	  3: l2->l2, Arg_1: 0 {O(1)}
4.13/2.01	
4.13/2.01	`Upper:
4.13/2.01	
4.13/2.01	  0: l0->l1, Arg_0: Arg_0 {O(n)}
4.13/2.01	
4.13/2.01	  0: l0->l1, Arg_1: Arg_1 {O(n)}
4.13/2.01	
4.13/2.01	  1: l1->l1, Arg_0: Arg_0 {O(n)}
4.13/2.01	
4.13/2.01	  1: l1->l1, Arg_1: max([Arg_0, max([Arg_0, Arg_1])])+max([0, Arg_0*Arg_0]) {O(n^2)}
4.13/2.01	
4.13/2.01	  2: l1->l2, Arg_0: 0 {O(1)}
4.13/2.01	
4.13/2.01	  2: l1->l2, Arg_1: max([Arg_1, max([Arg_0, max([Arg_0, Arg_1])])+max([0, Arg_0*Arg_0])]) {O(n^2)}
4.13/2.01	
4.13/2.01	  3: l2->l2, Arg_0: 0 {O(1)}
4.13/2.01	
4.13/2.01	  3: l2->l2, Arg_1: max([Arg_1, max([Arg_0, max([Arg_0, Arg_1])])+max([0, Arg_0*Arg_0])]) {O(n^2)}
4.13/2.01	
4.13/2.01	
4.13/2.01	----------------------------------------
4.13/2.01	
4.13/2.01	(2)
4.13/2.01	BOUNDS(1, max(max(Arg_1, Arg_0) + nat(Arg_0^2), Arg_1, 0) + max(2, 2 + Arg_0))
4.13/2.01	
4.13/2.01	----------------------------------------
4.13/2.01	
4.13/2.01	(3) Loat Proof (FINISHED)
4.13/2.01	
4.13/2.01	
4.13/2.01	### Pre-processing the ITS problem ###
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Initial linear ITS problem
4.13/2.01	
4.13/2.01	   Start location: l0
4.13/2.01	
4.13/2.01	      0: l0 -> l1 : [], cost: 1
4.13/2.01	
4.13/2.01	      1: l1 -> l1 : A'=-1+A, B'=A+B, [ A>=1 ], cost: 1
4.13/2.01	
4.13/2.01	      2: l1 -> l2 : [ 0>=A ], cost: 1
4.13/2.01	
4.13/2.01	      3: l2 -> l2 : B'=-1+B, [ B>=1 ], cost: 1
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	### Simplification by acceleration and chaining ###
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Accelerating simple loops of location 1.
4.13/2.01	
4.13/2.01	   Accelerating the following rules:
4.13/2.01	
4.13/2.01	      1: l1 -> l1 : A'=-1+A, B'=A+B, [ A>=1 ], cost: 1
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	   Accelerated rule 1 with metering function A, yielding the new rule 4.
4.13/2.01	
4.13/2.01	   Removing the simple loops: 1.
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Accelerating simple loops of location 2.
4.13/2.01	
4.13/2.01	   Accelerating the following rules:
4.13/2.01	
4.13/2.01	      3: l2 -> l2 : B'=-1+B, [ B>=1 ], cost: 1
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	   Accelerated rule 3 with metering function B, yielding the new rule 5.
4.13/2.01	
4.13/2.01	   Removing the simple loops: 3.
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Accelerated all simple loops using metering functions (where possible):
4.13/2.01	
4.13/2.01	   Start location: l0
4.13/2.01	
4.13/2.01	      0: l0 -> l1 : [], cost: 1
4.13/2.01	
4.13/2.01	      2: l1 -> l2 : [ 0>=A ], cost: 1
4.13/2.01	
4.13/2.01	      4: l1 -> l1 : A'=0, B'=1-1/2*A+1/2*A^2+B, [ A>=1 ], cost: A
4.13/2.01	
4.13/2.01	      5: l2 -> l2 : B'=0, [ B>=1 ], cost: B
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Chained accelerated rules (with incoming rules):
4.13/2.01	
4.13/2.01	   Start location: l0
4.13/2.01	
4.13/2.01	      0: l0 -> l1 : [], cost: 1
4.13/2.01	
4.13/2.01	      6: l0 -> l1 : A'=0, B'=1-1/2*A+1/2*A^2+B, [ A>=1 ], cost: 1+A
4.13/2.01	
4.13/2.01	      2: l1 -> l2 : [ 0>=A ], cost: 1
4.13/2.01	
4.13/2.01	      7: l1 -> l2 : B'=0, [ 0>=A && B>=1 ], cost: 1+B
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Removed unreachable locations (and leaf rules with constant cost):
4.13/2.01	
4.13/2.01	   Start location: l0
4.13/2.01	
4.13/2.01	      0: l0 -> l1 : [], cost: 1
4.13/2.01	
4.13/2.01	      6: l0 -> l1 : A'=0, B'=1-1/2*A+1/2*A^2+B, [ A>=1 ], cost: 1+A
4.13/2.01	
4.13/2.01	      7: l1 -> l2 : B'=0, [ 0>=A && B>=1 ], cost: 1+B
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Eliminated locations (on tree-shaped paths):
4.13/2.01	
4.13/2.01	   Start location: l0
4.13/2.01	
4.13/2.01	      8: l0 -> l2 : B'=0, [ 0>=A && B>=1 ], cost: 2+B
4.13/2.01	
4.13/2.01	      9: l0 -> l2 : A'=0, B'=0, [ A>=1 && 1-1/2*A+1/2*A^2+B>=1 ], cost: 3+1/2*A+1/2*A^2+B
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	### Computing asymptotic complexity ###
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Fully simplified ITS problem
4.13/2.01	
4.13/2.01	   Start location: l0
4.13/2.01	
4.13/2.01	      8: l0 -> l2 : B'=0, [ 0>=A && B>=1 ], cost: 2+B
4.13/2.01	
4.13/2.01	      9: l0 -> l2 : A'=0, B'=0, [ A>=1 && 1-1/2*A+1/2*A^2+B>=1 ], cost: 3+1/2*A+1/2*A^2+B
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Computing asymptotic complexity for rule 8
4.13/2.01	
4.13/2.01	   Solved the limit problem by the following transformations:
4.13/2.01	
4.13/2.01	      Created initial limit problem:
4.13/2.01	
4.13/2.01	      2+B (+), 1-A (+/+!), B (+/+!) [not solved]
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	      removing all constraints (solved by SMT)
4.13/2.01	
4.13/2.01	      resulting limit problem: [solved]
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	      applying transformation rule (C) using substitution {A==-n,B==n}
4.13/2.01	
4.13/2.01	      resulting limit problem:
4.13/2.01	
4.13/2.01	      [solved]
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	   Solution:
4.13/2.01	
4.13/2.01	      A / -n
4.13/2.01	
4.13/2.01	      B / n
4.13/2.01	
4.13/2.01	   Resulting cost 2+n has complexity: Poly(n^1)
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Found new complexity Poly(n^1).
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Computing asymptotic complexity for rule 9
4.13/2.01	
4.13/2.01	   Solved the limit problem by the following transformations:
4.13/2.01	
4.13/2.01	      Created initial limit problem:
4.13/2.01	
4.13/2.01	      3+1/2*A+1/2*A^2+B (+), A (+/+!), 1-1/2*A+1/2*A^2+B (+/+!) [not solved]
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	      applying transformation rule (C) using substitution {A==1}
4.13/2.01	
4.13/2.01	      resulting limit problem:
4.13/2.01	
4.13/2.01	      1 (+/+!), 1+B (+/+!), 4+B (+) [not solved]
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	      applying transformation rule (B), deleting 1 (+/+!)
4.13/2.01	
4.13/2.01	      resulting limit problem:
4.13/2.01	
4.13/2.01	      1+B (+/+!), 4+B (+) [not solved]
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	      removing all constraints (solved by SMT)
4.13/2.01	
4.13/2.01	      resulting limit problem: [solved]
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	      applying transformation rule (C) using substitution {B==n}
4.13/2.01	
4.13/2.01	      resulting limit problem:
4.13/2.01	
4.13/2.01	      [solved]
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	   Solved the limit problem by the following transformations:
4.13/2.01	
4.13/2.01	      Created initial limit problem:
4.13/2.01	
4.13/2.01	      3+1/2*A+1/2*A^2+B (+), A (+/+!), 1-1/2*A+1/2*A^2+B (+/+!) [not solved]
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	      removing all constraints (solved by SMT)
4.13/2.01	
4.13/2.01	      resulting limit problem: [solved]
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	      applying transformation rule (C) using substitution {A==n,B==0}
4.13/2.01	
4.13/2.01	      resulting limit problem:
4.13/2.01	
4.13/2.01	      [solved]
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	   Solution:
4.13/2.01	
4.13/2.01	      A / n
4.13/2.01	
4.13/2.01	      B / 0
4.13/2.01	
4.13/2.01	   Resulting cost 3+1/2*n^2+1/2*n has complexity: Poly(n^2)
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Found new complexity Poly(n^2).
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	Obtained the following overall complexity (w.r.t. the length of the input n):
4.13/2.01	
4.13/2.01	   Complexity:  Poly(n^2)
4.13/2.01	
4.13/2.01	   Cpx degree:  2
4.13/2.01	
4.13/2.01	   Solved cost: 3+1/2*n^2+1/2*n
4.13/2.01	
4.13/2.01	   Rule cost:   3+1/2*A+1/2*A^2+B
4.13/2.01	
4.13/2.01	   Rule guard:  [ A>=1 && 1-1/2*A+1/2*A^2+B>=1 ]
4.13/2.01	
4.13/2.01	
4.13/2.01	
4.13/2.01	WORST_CASE(Omega(n^2),?)
4.13/2.01	
4.13/2.01	
4.13/2.01	----------------------------------------
4.13/2.01	
4.13/2.01	(4)
4.13/2.01	BOUNDS(n^2, INF)
4.13/2.04	EOF