3.72/1.78	WORST_CASE(Omega(n^1), O(n^1))
3.78/1.79	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.78/1.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.78/1.79	
3.78/1.79	
3.78/1.79	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
3.78/1.79	
3.78/1.79	(0) CpxIntTrs
3.78/1.79	(1) Koat Proof [FINISHED, 15 ms]
3.78/1.79	(2) BOUNDS(1, n^1)
3.78/1.79	(3) Loat Proof [FINISHED, 121 ms]
3.78/1.79	(4) BOUNDS(n^1, INF)
3.78/1.79	
3.78/1.79	
3.78/1.79	----------------------------------------
3.78/1.79	
3.78/1.79	(0)
3.78/1.79	Obligation:
3.78/1.79	Complexity Int TRS consisting of the following rules:
3.78/1.79	f2(A, B) -> Com_1(f2(-(1) + A, B)) :|: A >= 2
3.78/1.79	f2(A, B) -> Com_1(f1(-(1) + A, C)) :|: 1 >= A
3.78/1.79	f300(A, B) -> Com_1(f2(A, B)) :|: TRUE
3.78/1.79	
3.78/1.79	The start-symbols are:[f300_2]
3.78/1.79	
3.78/1.79	
3.78/1.79	----------------------------------------
3.78/1.79	
3.78/1.79	(1) Koat Proof (FINISHED)
3.78/1.79	YES(?, ar_0 + 2)
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Initial complexity problem:
3.78/1.79	
3.78/1.79	1:	T:
3.78/1.79	
3.78/1.79			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ ar_0 >= 2 ]
3.78/1.79	
3.78/1.79			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f1(ar_0 - 1, c)) [ 1 >= ar_0 ]
3.78/1.79	
3.78/1.79			(Comp: ?, Cost: 1)    f300(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1))
3.78/1.79	
3.78/1.79			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f300(ar_0, ar_1)) [ 0 <= 0 ]
3.78/1.79	
3.78/1.79		start location:	koat_start
3.78/1.79	
3.78/1.79		leaf cost:	0
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.78/1.79	
3.78/1.79	2:	T:
3.78/1.79	
3.78/1.79			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ ar_0 >= 2 ]
3.78/1.79	
3.78/1.79			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f1(ar_0 - 1, c)) [ 1 >= ar_0 ]
3.78/1.79	
3.78/1.79			(Comp: 1, Cost: 1)    f300(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1))
3.78/1.79	
3.78/1.79			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f300(ar_0, ar_1)) [ 0 <= 0 ]
3.78/1.79	
3.78/1.79		start location:	koat_start
3.78/1.79	
3.78/1.79		leaf cost:	0
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	A polynomial rank function with
3.78/1.79	
3.78/1.79		Pol(f2) = 1
3.78/1.79	
3.78/1.79		Pol(f1) = 0
3.78/1.79	
3.78/1.79		Pol(f300) = 1
3.78/1.79	
3.78/1.79		Pol(koat_start) = 1
3.78/1.79	
3.78/1.79	orients all transitions weakly and the transition
3.78/1.79	
3.78/1.79		f2(ar_0, ar_1) -> Com_1(f1(ar_0 - 1, c)) [ 1 >= ar_0 ]
3.78/1.79	
3.78/1.79	strictly and produces the following problem:
3.78/1.79	
3.78/1.79	3:	T:
3.78/1.79	
3.78/1.79			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ ar_0 >= 2 ]
3.78/1.79	
3.78/1.79			(Comp: 1, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f1(ar_0 - 1, c)) [ 1 >= ar_0 ]
3.78/1.79	
3.78/1.79			(Comp: 1, Cost: 1)    f300(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1))
3.78/1.79	
3.78/1.79			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f300(ar_0, ar_1)) [ 0 <= 0 ]
3.78/1.79	
3.78/1.79		start location:	koat_start
3.78/1.79	
3.78/1.79		leaf cost:	0
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	A polynomial rank function with
3.78/1.79	
3.78/1.79		Pol(f2) = V_1
3.78/1.79	
3.78/1.79		Pol(f1) = V_1
3.78/1.79	
3.78/1.79		Pol(f300) = V_1
3.78/1.79	
3.78/1.79		Pol(koat_start) = V_1
3.78/1.79	
3.78/1.79	orients all transitions weakly and the transition
3.78/1.79	
3.78/1.79		f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ ar_0 >= 2 ]
3.78/1.79	
3.78/1.79	strictly and produces the following problem:
3.78/1.79	
3.78/1.79	4:	T:
3.78/1.79	
3.78/1.79			(Comp: ar_0, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ ar_0 >= 2 ]
3.78/1.79	
3.78/1.79			(Comp: 1, Cost: 1)       f2(ar_0, ar_1) -> Com_1(f1(ar_0 - 1, c)) [ 1 >= ar_0 ]
3.78/1.79	
3.78/1.79			(Comp: 1, Cost: 1)       f300(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1))
3.78/1.79	
3.78/1.79			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1) -> Com_1(f300(ar_0, ar_1)) [ 0 <= 0 ]
3.78/1.79	
3.78/1.79		start location:	koat_start
3.78/1.79	
3.78/1.79		leaf cost:	0
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Complexity upper bound ar_0 + 2
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Time: 0.041 sec (SMT: 0.039 sec)
3.78/1.79	
3.78/1.79	
3.78/1.79	----------------------------------------
3.78/1.79	
3.78/1.79	(2)
3.78/1.79	BOUNDS(1, n^1)
3.78/1.79	
3.78/1.79	----------------------------------------
3.78/1.79	
3.78/1.79	(3) Loat Proof (FINISHED)
3.78/1.79	
3.78/1.79	
3.78/1.79	### Pre-processing the ITS problem ###
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Initial linear ITS problem
3.78/1.79	
3.78/1.79	   Start location: f300
3.78/1.79	
3.78/1.79	      0: f2 -> f2 : A'=-1+A, [ A>=2 ], cost: 1
3.78/1.79	
3.78/1.79	      1: f2 -> f1 : A'=-1+A, B'=free, [ 1>=A ], cost: 1
3.78/1.79	
3.78/1.79	      2: f300 -> f2 : [], cost: 1
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Removed unreachable and leaf rules:
3.78/1.79	
3.78/1.79	   Start location: f300
3.78/1.79	
3.78/1.79	      0: f2 -> f2 : A'=-1+A, [ A>=2 ], cost: 1
3.78/1.79	
3.78/1.79	      2: f300 -> f2 : [], cost: 1
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	### Simplification by acceleration and chaining ###
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Accelerating simple loops of location 0.
3.78/1.79	
3.78/1.79	   Accelerating the following rules:
3.78/1.79	
3.78/1.79	      0: f2 -> f2 : A'=-1+A, [ A>=2 ], cost: 1
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	   Accelerated rule 0 with metering function -1+A, yielding the new rule 3.
3.78/1.79	
3.78/1.79	   Removing the simple loops: 0.
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Accelerated all simple loops using metering functions (where possible):
3.78/1.79	
3.78/1.79	   Start location: f300
3.78/1.79	
3.78/1.79	      3: f2 -> f2 : A'=1, [ A>=2 ], cost: -1+A
3.78/1.79	
3.78/1.79	      2: f300 -> f2 : [], cost: 1
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Chained accelerated rules (with incoming rules):
3.78/1.79	
3.78/1.79	   Start location: f300
3.78/1.79	
3.78/1.79	      2: f300 -> f2 : [], cost: 1
3.78/1.79	
3.78/1.79	      4: f300 -> f2 : A'=1, [ A>=2 ], cost: A
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Removed unreachable locations (and leaf rules with constant cost):
3.78/1.79	
3.78/1.79	   Start location: f300
3.78/1.79	
3.78/1.79	      4: f300 -> f2 : A'=1, [ A>=2 ], cost: A
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	### Computing asymptotic complexity ###
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Fully simplified ITS problem
3.78/1.79	
3.78/1.79	   Start location: f300
3.78/1.79	
3.78/1.79	      4: f300 -> f2 : A'=1, [ A>=2 ], cost: A
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Computing asymptotic complexity for rule 4
3.78/1.79	
3.78/1.79	   Solved the limit problem by the following transformations:
3.78/1.79	
3.78/1.79	      Created initial limit problem:
3.78/1.79	
3.78/1.79	      -1+A (+/+!), A (+) [not solved]
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	      removing all constraints (solved by SMT)
3.78/1.79	
3.78/1.79	      resulting limit problem: [solved]
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	      applying transformation rule (C) using substitution {A==n}
3.78/1.79	
3.78/1.79	      resulting limit problem:
3.78/1.79	
3.78/1.79	      [solved]
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	   Solution:
3.78/1.79	
3.78/1.79	      A / n
3.78/1.79	
3.78/1.79	   Resulting cost n has complexity: Poly(n^1)
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Found new complexity Poly(n^1).
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	Obtained the following overall complexity (w.r.t. the length of the input n):
3.78/1.79	
3.78/1.79	   Complexity:  Poly(n^1)
3.78/1.79	
3.78/1.79	   Cpx degree:  1
3.78/1.79	
3.78/1.79	   Solved cost: n
3.78/1.79	
3.78/1.79	   Rule cost:   A
3.78/1.79	
3.78/1.79	   Rule guard:  [ A>=2 ]
3.78/1.79	
3.78/1.79	
3.78/1.79	
3.78/1.79	WORST_CASE(Omega(n^1),?)
3.78/1.79	
3.78/1.79	
3.78/1.79	----------------------------------------
3.78/1.79	
3.78/1.79	(4)
3.78/1.79	BOUNDS(n^1, INF)
3.78/1.81	EOF