4.35/2.16	WORST_CASE(Omega(n^1), O(n^1))
4.54/2.17	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
4.54/2.17	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.54/2.17	
4.54/2.17	
4.54/2.17	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
4.54/2.17	
4.54/2.17	(0) CpxIntTrs
4.54/2.17	(1) Koat Proof [FINISHED, 127 ms]
4.54/2.17	(2) BOUNDS(1, n^1)
4.54/2.17	(3) Loat Proof [FINISHED, 531 ms]
4.54/2.17	(4) BOUNDS(n^1, INF)
4.54/2.17	
4.54/2.17	
4.54/2.17	----------------------------------------
4.54/2.17	
4.54/2.17	(0)
4.54/2.17	Obligation:
4.54/2.17	Complexity Int TRS consisting of the following rules:
4.54/2.17	f2(A, B, C, D, E) -> Com_1(f1(A, F, C, D, E)) :|: 1 >= A
4.54/2.17	f2(A, B, C, D, E) -> Com_1(f300(A, B, C, D, E)) :|: A >= 2 && C >= 2
4.54/2.17	f300(A, B, C, D, E) -> Com_1(f1(A, F, C, D, E)) :|: D + 1 >= 0
4.54/2.17	f300(A, B, C, D, E) -> Com_1(f300(A, B, C, 1 + D, 1 + E)) :|: E >= 0 && 0 >= 2 + D
4.54/2.17	f300(A, B, C, D, E) -> Com_1(f300(A, B, C, 1 + D, 1 + E)) :|: 0 >= 2 + E && 0 >= 2 + D
4.54/2.17	
4.54/2.17	The start-symbols are:[f2_5]
4.54/2.17	
4.54/2.17	
4.54/2.17	----------------------------------------
4.54/2.17	
4.54/2.17	(1) Koat Proof (FINISHED)
4.54/2.17	YES(?, 2*ar_3 + 3)
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Initial complexity problem:
4.54/2.17	
4.54/2.17	1:	T:
4.54/2.17	
4.54/2.17			(Comp: ?, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f1(ar_0, f, ar_2, ar_3, ar_4)) [ 1 >= ar_0 ]
4.54/2.17	
4.54/2.17			(Comp: ?, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= 2 /\ ar_2 >= 2 ]
4.54/2.17	
4.54/2.17			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f1(ar_0, f, ar_2, ar_3, ar_4)) [ ar_3 + 1 >= 0 ]
4.54/2.17	
4.54/2.17			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3 + 1, ar_4 + 1)) [ ar_4 >= 0 /\ 0 >= ar_3 + 2 ]
4.54/2.17	
4.54/2.17			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3 + 1, ar_4 + 1)) [ 0 >= ar_4 + 2 /\ 0 >= ar_3 + 2 ]
4.54/2.17	
4.54/2.17			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f2(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ]
4.54/2.17	
4.54/2.17		start location:	koat_start
4.54/2.17	
4.54/2.17		leaf cost:	0
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Repeatedly propagating knowledge in problem 1 produces the following problem:
4.54/2.17	
4.54/2.17	2:	T:
4.54/2.17	
4.54/2.17			(Comp: 1, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f1(ar_0, f, ar_2, ar_3, ar_4)) [ 1 >= ar_0 ]
4.54/2.17	
4.54/2.17			(Comp: 1, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= 2 /\ ar_2 >= 2 ]
4.54/2.17	
4.54/2.17			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f1(ar_0, f, ar_2, ar_3, ar_4)) [ ar_3 + 1 >= 0 ]
4.54/2.17	
4.54/2.17			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3 + 1, ar_4 + 1)) [ ar_4 >= 0 /\ 0 >= ar_3 + 2 ]
4.54/2.17	
4.54/2.17			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3 + 1, ar_4 + 1)) [ 0 >= ar_4 + 2 /\ 0 >= ar_3 + 2 ]
4.54/2.17	
4.54/2.17			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f2(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ]
4.54/2.17	
4.54/2.17		start location:	koat_start
4.54/2.17	
4.54/2.17		leaf cost:	0
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	A polynomial rank function with
4.54/2.17	
4.54/2.17		Pol(f2) = 1
4.54/2.17	
4.54/2.17		Pol(f1) = 0
4.54/2.17	
4.54/2.17		Pol(f300) = 1
4.54/2.17	
4.54/2.17		Pol(koat_start) = 1
4.54/2.17	
4.54/2.17	orients all transitions weakly and the transition
4.54/2.17	
4.54/2.17		f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f1(ar_0, f, ar_2, ar_3, ar_4)) [ ar_3 + 1 >= 0 ]
4.54/2.17	
4.54/2.17	strictly and produces the following problem:
4.54/2.17	
4.54/2.17	3:	T:
4.54/2.17	
4.54/2.17			(Comp: 1, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f1(ar_0, f, ar_2, ar_3, ar_4)) [ 1 >= ar_0 ]
4.54/2.17	
4.54/2.17			(Comp: 1, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= 2 /\ ar_2 >= 2 ]
4.54/2.17	
4.54/2.17			(Comp: 1, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f1(ar_0, f, ar_2, ar_3, ar_4)) [ ar_3 + 1 >= 0 ]
4.54/2.17	
4.54/2.17			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3 + 1, ar_4 + 1)) [ ar_4 >= 0 /\ 0 >= ar_3 + 2 ]
4.54/2.17	
4.54/2.17			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3 + 1, ar_4 + 1)) [ 0 >= ar_4 + 2 /\ 0 >= ar_3 + 2 ]
4.54/2.17	
4.54/2.17			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f2(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ]
4.54/2.17	
4.54/2.17		start location:	koat_start
4.54/2.17	
4.54/2.17		leaf cost:	0
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	A polynomial rank function with
4.54/2.17	
4.54/2.17		Pol(f2) = -V_4
4.54/2.17	
4.54/2.17		Pol(f1) = -V_4
4.54/2.17	
4.54/2.17		Pol(f300) = -V_4
4.54/2.17	
4.54/2.17		Pol(koat_start) = -V_4
4.54/2.17	
4.54/2.17	orients all transitions weakly and the transitions
4.54/2.17	
4.54/2.17		f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3 + 1, ar_4 + 1)) [ ar_4 >= 0 /\ 0 >= ar_3 + 2 ]
4.54/2.17	
4.54/2.17		f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3 + 1, ar_4 + 1)) [ 0 >= ar_4 + 2 /\ 0 >= ar_3 + 2 ]
4.54/2.17	
4.54/2.17	strictly and produces the following problem:
4.54/2.17	
4.54/2.17	4:	T:
4.54/2.17	
4.54/2.17			(Comp: 1, Cost: 1)       f2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f1(ar_0, f, ar_2, ar_3, ar_4)) [ 1 >= ar_0 ]
4.54/2.17	
4.54/2.17			(Comp: 1, Cost: 1)       f2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= 2 /\ ar_2 >= 2 ]
4.54/2.17	
4.54/2.17			(Comp: 1, Cost: 1)       f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f1(ar_0, f, ar_2, ar_3, ar_4)) [ ar_3 + 1 >= 0 ]
4.54/2.17	
4.54/2.17			(Comp: ar_3, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3 + 1, ar_4 + 1)) [ ar_4 >= 0 /\ 0 >= ar_3 + 2 ]
4.54/2.17	
4.54/2.17			(Comp: ar_3, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3 + 1, ar_4 + 1)) [ 0 >= ar_4 + 2 /\ 0 >= ar_3 + 2 ]
4.54/2.17	
4.54/2.17			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f2(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ]
4.54/2.17	
4.54/2.17		start location:	koat_start
4.54/2.17	
4.54/2.17		leaf cost:	0
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Complexity upper bound 2*ar_3 + 3
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Time: 0.115 sec (SMT: 0.108 sec)
4.54/2.17	
4.54/2.17	
4.54/2.17	----------------------------------------
4.54/2.17	
4.54/2.17	(2)
4.54/2.17	BOUNDS(1, n^1)
4.54/2.17	
4.54/2.17	----------------------------------------
4.54/2.17	
4.54/2.17	(3) Loat Proof (FINISHED)
4.54/2.17	
4.54/2.17	
4.54/2.17	### Pre-processing the ITS problem ###
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Initial linear ITS problem
4.54/2.17	
4.54/2.17	   Start location: f2
4.54/2.17	
4.54/2.17	      0: f2 -> f1 : B'=free, [ 1>=A ], cost: 1
4.54/2.17	
4.54/2.17	      1: f2 -> f300 : [ A>=2 && C>=2 ], cost: 1
4.54/2.17	
4.54/2.17	      2: f300 -> f1 : B'=free_1, [ 1+D>=0 ], cost: 1
4.54/2.17	
4.54/2.17	      3: f300 -> f300 : D'=1+D, E'=1+E, [ E>=0 && 0>=2+D ], cost: 1
4.54/2.17	
4.54/2.17	      4: f300 -> f300 : D'=1+D, E'=1+E, [ 0>=2+E && 0>=2+D ], cost: 1
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Removed unreachable and leaf rules:
4.54/2.17	
4.54/2.17	   Start location: f2
4.54/2.17	
4.54/2.17	      1: f2 -> f300 : [ A>=2 && C>=2 ], cost: 1
4.54/2.17	
4.54/2.17	      3: f300 -> f300 : D'=1+D, E'=1+E, [ E>=0 && 0>=2+D ], cost: 1
4.54/2.17	
4.54/2.17	      4: f300 -> f300 : D'=1+D, E'=1+E, [ 0>=2+E && 0>=2+D ], cost: 1
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	### Simplification by acceleration and chaining ###
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Accelerating simple loops of location 1.
4.54/2.17	
4.54/2.17	   Accelerating the following rules:
4.54/2.17	
4.54/2.17	      3: f300 -> f300 : D'=1+D, E'=1+E, [ E>=0 && 0>=2+D ], cost: 1
4.54/2.17	
4.54/2.17	      4: f300 -> f300 : D'=1+D, E'=1+E, [ 0>=2+E && 0>=2+D ], cost: 1
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	   Accelerated rule 3 with metering function -1-D, yielding the new rule 5.
4.54/2.17	
4.54/2.17	   Accelerated rule 4 with metering function -1-D (after adding D>=E), yielding the new rule 6.
4.54/2.17	
4.54/2.17	   Accelerated rule 4 with metering function -1-E (after adding D<=E), yielding the new rule 7.
4.54/2.17	
4.54/2.17	   Removing the simple loops: 3 4.
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Accelerated all simple loops using metering functions (where possible):
4.54/2.17	
4.54/2.17	   Start location: f2
4.54/2.17	
4.54/2.17	      1: f2 -> f300 : [ A>=2 && C>=2 ], cost: 1
4.54/2.17	
4.54/2.17	      5: f300 -> f300 : D'=-1, E'=-1-D+E, [ E>=0 && 0>=2+D ], cost: -1-D
4.54/2.17	
4.54/2.17	      6: f300 -> f300 : D'=-1, E'=-1-D+E, [ 0>=2+E && 0>=2+D && D>=E ], cost: -1-D
4.54/2.17	
4.54/2.17	      7: f300 -> f300 : D'=-1+D-E, E'=-1, [ 0>=2+E && 0>=2+D && D<=E ], cost: -1-E
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Chained accelerated rules (with incoming rules):
4.54/2.17	
4.54/2.17	   Start location: f2
4.54/2.17	
4.54/2.17	      1: f2 -> f300 : [ A>=2 && C>=2 ], cost: 1
4.54/2.17	
4.54/2.17	      8: f2 -> f300 : D'=-1, E'=-1-D+E, [ A>=2 && C>=2 && E>=0 && 0>=2+D ], cost: -D
4.54/2.17	
4.54/2.17	      9: f2 -> f300 : D'=-1, E'=-1-D+E, [ A>=2 && C>=2 && 0>=2+E && 0>=2+D && D>=E ], cost: -D
4.54/2.17	
4.54/2.17	     10: f2 -> f300 : D'=-1+D-E, E'=-1, [ A>=2 && C>=2 && 0>=2+E && 0>=2+D && D<=E ], cost: -E
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Removed unreachable locations (and leaf rules with constant cost):
4.54/2.17	
4.54/2.17	   Start location: f2
4.54/2.17	
4.54/2.17	      8: f2 -> f300 : D'=-1, E'=-1-D+E, [ A>=2 && C>=2 && E>=0 && 0>=2+D ], cost: -D
4.54/2.17	
4.54/2.17	      9: f2 -> f300 : D'=-1, E'=-1-D+E, [ A>=2 && C>=2 && 0>=2+E && 0>=2+D && D>=E ], cost: -D
4.54/2.17	
4.54/2.17	     10: f2 -> f300 : D'=-1+D-E, E'=-1, [ A>=2 && C>=2 && 0>=2+E && 0>=2+D && D<=E ], cost: -E
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	### Computing asymptotic complexity ###
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Fully simplified ITS problem
4.54/2.17	
4.54/2.17	   Start location: f2
4.54/2.17	
4.54/2.17	      8: f2 -> f300 : D'=-1, E'=-1-D+E, [ A>=2 && C>=2 && E>=0 && 0>=2+D ], cost: -D
4.54/2.17	
4.54/2.17	      9: f2 -> f300 : D'=-1, E'=-1-D+E, [ A>=2 && C>=2 && 0>=2+E && 0>=2+D && D>=E ], cost: -D
4.54/2.17	
4.54/2.17	     10: f2 -> f300 : D'=-1+D-E, E'=-1, [ A>=2 && C>=2 && 0>=2+E && 0>=2+D && D<=E ], cost: -E
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Computing asymptotic complexity for rule 8
4.54/2.17	
4.54/2.17	   Solved the limit problem by the following transformations:
4.54/2.17	
4.54/2.17	      Created initial limit problem:
4.54/2.17	
4.54/2.17	      -1+A (+/+!), 1+E (+/+!), -1-D (+/+!), -D (+), -1+C (+/+!) [not solved]
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	      applying transformation rule (C) using substitution {A==2}
4.54/2.17	
4.54/2.17	      resulting limit problem:
4.54/2.17	
4.54/2.17	      1 (+/+!), 1+E (+/+!), -1-D (+/+!), -D (+), -1+C (+/+!) [not solved]
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	      applying transformation rule (C) using substitution {C==2}
4.54/2.17	
4.54/2.17	      resulting limit problem:
4.54/2.17	
4.54/2.17	      1 (+/+!), 1+E (+/+!), -1-D (+/+!), -D (+) [not solved]
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	      applying transformation rule (C) using substitution {E==0}
4.54/2.17	
4.54/2.17	      resulting limit problem:
4.54/2.17	
4.54/2.17	      1 (+/+!), -1-D (+/+!), -D (+) [not solved]
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	      applying transformation rule (B), deleting 1 (+/+!)
4.54/2.17	
4.54/2.17	      resulting limit problem:
4.54/2.17	
4.54/2.17	      -1-D (+/+!), -D (+) [not solved]
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	      removing all constraints (solved by SMT)
4.54/2.17	
4.54/2.17	      resulting limit problem: [solved]
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	      applying transformation rule (C) using substitution {D==-n}
4.54/2.17	
4.54/2.17	      resulting limit problem:
4.54/2.17	
4.54/2.17	      [solved]
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	   Solution:
4.54/2.17	
4.54/2.17	      C / 2
4.54/2.17	
4.54/2.17	      D / -n
4.54/2.17	
4.54/2.17	      A / 2
4.54/2.17	
4.54/2.17	      E / 0
4.54/2.17	
4.54/2.17	   Resulting cost n has complexity: Poly(n^1)
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Found new complexity Poly(n^1).
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	Obtained the following overall complexity (w.r.t. the length of the input n):
4.54/2.17	
4.54/2.17	   Complexity:  Poly(n^1)
4.54/2.17	
4.54/2.17	   Cpx degree:  1
4.54/2.17	
4.54/2.17	   Solved cost: n
4.54/2.17	
4.54/2.17	   Rule cost:   -D
4.54/2.17	
4.54/2.17	   Rule guard:  [ A>=2 && C>=2 && E>=0 && 0>=2+D ]
4.54/2.17	
4.54/2.17	
4.54/2.17	
4.54/2.17	WORST_CASE(Omega(n^1),?)
4.54/2.17	
4.54/2.17	
4.54/2.17	----------------------------------------
4.54/2.17	
4.54/2.17	(4)
4.54/2.17	BOUNDS(n^1, INF)
4.56/2.20	EOF