3.74/1.81	WORST_CASE(Omega(n^1), O(n^1))
3.88/1.81	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.88/1.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.88/1.81	
3.88/1.81	
3.88/1.81	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
3.88/1.81	
3.88/1.81	(0) CpxIntTrs
3.88/1.81	(1) Koat Proof [FINISHED, 78 ms]
3.88/1.81	(2) BOUNDS(1, n^1)
3.88/1.81	(3) Loat Proof [FINISHED, 144 ms]
3.88/1.81	(4) BOUNDS(n^1, INF)
3.88/1.81	
3.88/1.81	
3.88/1.81	----------------------------------------
3.88/1.81	
3.88/1.81	(0)
3.88/1.81	Obligation:
3.88/1.81	Complexity Int TRS consisting of the following rules:
3.88/1.81	l0(A, B) -> Com_1(l1(0, B)) :|: TRUE
3.88/1.81	l1(A, B) -> Com_1(l1(A + 1, B - 1)) :|: B >= 1
3.88/1.81	l1(A, B) -> Com_1(l2(A, B)) :|: 0 >= B
3.88/1.81	
3.88/1.81	The start-symbols are:[l0_2]
3.88/1.81	
3.88/1.81	
3.88/1.81	----------------------------------------
3.88/1.81	
3.88/1.81	(1) Koat Proof (FINISHED)
3.88/1.81	YES(?, ar_1 + 2)
3.88/1.81	
3.88/1.81	
3.88/1.81	
3.88/1.81	Initial complexity problem:
3.88/1.81	
3.88/1.81	1:	T:
3.88/1.81	
3.88/1.81			(Comp: ?, Cost: 1)    l0(ar_0, ar_1) -> Com_1(l1(0, ar_1))
3.88/1.81	
3.88/1.81			(Comp: ?, Cost: 1)    l1(ar_0, ar_1) -> Com_1(l1(ar_0 + 1, ar_1 - 1)) [ ar_1 >= 1 ]
3.88/1.81	
3.88/1.81			(Comp: ?, Cost: 1)    l1(ar_0, ar_1) -> Com_1(l2(ar_0, ar_1)) [ 0 >= ar_1 ]
3.88/1.81	
3.88/1.81			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(l0(ar_0, ar_1)) [ 0 <= 0 ]
3.88/1.81	
3.88/1.81		start location:	koat_start
3.88/1.81	
3.88/1.81		leaf cost:	0
3.88/1.81	
3.88/1.81	
3.88/1.81	
3.88/1.81	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.88/1.81	
3.88/1.81	2:	T:
3.88/1.81	
3.88/1.81			(Comp: 1, Cost: 1)    l0(ar_0, ar_1) -> Com_1(l1(0, ar_1))
3.88/1.81	
3.88/1.81			(Comp: ?, Cost: 1)    l1(ar_0, ar_1) -> Com_1(l1(ar_0 + 1, ar_1 - 1)) [ ar_1 >= 1 ]
3.88/1.81	
3.88/1.81			(Comp: ?, Cost: 1)    l1(ar_0, ar_1) -> Com_1(l2(ar_0, ar_1)) [ 0 >= ar_1 ]
3.88/1.81	
3.88/1.81			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(l0(ar_0, ar_1)) [ 0 <= 0 ]
3.88/1.81	
3.88/1.81		start location:	koat_start
3.88/1.81	
3.88/1.81		leaf cost:	0
3.88/1.81	
3.88/1.81	
3.88/1.81	
3.88/1.81	A polynomial rank function with
3.88/1.81	
3.88/1.81		Pol(l0) = 1
3.88/1.81	
3.88/1.81		Pol(l1) = 1
3.88/1.81	
3.88/1.81		Pol(l2) = 0
3.88/1.81	
3.88/1.81		Pol(koat_start) = 1
3.88/1.81	
3.88/1.81	orients all transitions weakly and the transition
3.88/1.81	
3.88/1.81		l1(ar_0, ar_1) -> Com_1(l2(ar_0, ar_1)) [ 0 >= ar_1 ]
3.88/1.81	
3.88/1.81	strictly and produces the following problem:
3.88/1.81	
3.88/1.81	3:	T:
3.88/1.81	
3.88/1.81			(Comp: 1, Cost: 1)    l0(ar_0, ar_1) -> Com_1(l1(0, ar_1))
3.88/1.81	
3.88/1.81			(Comp: ?, Cost: 1)    l1(ar_0, ar_1) -> Com_1(l1(ar_0 + 1, ar_1 - 1)) [ ar_1 >= 1 ]
3.88/1.81	
3.88/1.81			(Comp: 1, Cost: 1)    l1(ar_0, ar_1) -> Com_1(l2(ar_0, ar_1)) [ 0 >= ar_1 ]
3.88/1.81	
3.88/1.81			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(l0(ar_0, ar_1)) [ 0 <= 0 ]
3.88/1.82	
3.88/1.82		start location:	koat_start
3.88/1.82	
3.88/1.82		leaf cost:	0
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	A polynomial rank function with
3.88/1.82	
3.88/1.82		Pol(l0) = V_2
3.88/1.82	
3.88/1.82		Pol(l1) = V_2
3.88/1.82	
3.88/1.82		Pol(l2) = V_2
3.88/1.82	
3.88/1.82		Pol(koat_start) = V_2
3.88/1.82	
3.88/1.82	orients all transitions weakly and the transition
3.88/1.82	
3.88/1.82		l1(ar_0, ar_1) -> Com_1(l1(ar_0 + 1, ar_1 - 1)) [ ar_1 >= 1 ]
3.88/1.82	
3.88/1.82	strictly and produces the following problem:
3.88/1.82	
3.88/1.82	4:	T:
3.88/1.82	
3.88/1.82			(Comp: 1, Cost: 1)       l0(ar_0, ar_1) -> Com_1(l1(0, ar_1))
3.88/1.82	
3.88/1.82			(Comp: ar_1, Cost: 1)    l1(ar_0, ar_1) -> Com_1(l1(ar_0 + 1, ar_1 - 1)) [ ar_1 >= 1 ]
3.88/1.82	
3.88/1.82			(Comp: 1, Cost: 1)       l1(ar_0, ar_1) -> Com_1(l2(ar_0, ar_1)) [ 0 >= ar_1 ]
3.88/1.82	
3.88/1.82			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1) -> Com_1(l0(ar_0, ar_1)) [ 0 <= 0 ]
3.88/1.82	
3.88/1.82		start location:	koat_start
3.88/1.82	
3.88/1.82		leaf cost:	0
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	Complexity upper bound ar_1 + 2
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	Time: 0.043 sec (SMT: 0.041 sec)
3.88/1.82	
3.88/1.82	
3.88/1.82	----------------------------------------
3.88/1.82	
3.88/1.82	(2)
3.88/1.82	BOUNDS(1, n^1)
3.88/1.82	
3.88/1.82	----------------------------------------
3.88/1.82	
3.88/1.82	(3) Loat Proof (FINISHED)
3.88/1.82	
3.88/1.82	
3.88/1.82	### Pre-processing the ITS problem ###
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	Initial linear ITS problem
3.88/1.82	
3.88/1.82	   Start location: l0
3.88/1.82	
3.88/1.82	      0: l0 -> l1 : A'=0, [], cost: 1
3.88/1.82	
3.88/1.82	      1: l1 -> l1 : A'=1+A, B'=-1+B, [ B>=1 ], cost: 1
3.88/1.82	
3.88/1.82	      2: l1 -> l2 : [ 0>=B ], cost: 1
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	Removed unreachable and leaf rules:
3.88/1.82	
3.88/1.82	   Start location: l0
3.88/1.82	
3.88/1.82	      0: l0 -> l1 : A'=0, [], cost: 1
3.88/1.82	
3.88/1.82	      1: l1 -> l1 : A'=1+A, B'=-1+B, [ B>=1 ], cost: 1
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	### Simplification by acceleration and chaining ###
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	Accelerating simple loops of location 1.
3.88/1.82	
3.88/1.82	   Accelerating the following rules:
3.88/1.82	
3.88/1.82	      1: l1 -> l1 : A'=1+A, B'=-1+B, [ B>=1 ], cost: 1
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	   Accelerated rule 1 with metering function B, yielding the new rule 3.
3.88/1.82	
3.88/1.82	   Removing the simple loops: 1.
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	Accelerated all simple loops using metering functions (where possible):
3.88/1.82	
3.88/1.82	   Start location: l0
3.88/1.82	
3.88/1.82	      0: l0 -> l1 : A'=0, [], cost: 1
3.88/1.82	
3.88/1.82	      3: l1 -> l1 : A'=A+B, B'=0, [ B>=1 ], cost: B
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	Chained accelerated rules (with incoming rules):
3.88/1.82	
3.88/1.82	   Start location: l0
3.88/1.82	
3.88/1.82	      0: l0 -> l1 : A'=0, [], cost: 1
3.88/1.82	
3.88/1.82	      4: l0 -> l1 : A'=B, B'=0, [ B>=1 ], cost: 1+B
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	Removed unreachable locations (and leaf rules with constant cost):
3.88/1.82	
3.88/1.82	   Start location: l0
3.88/1.82	
3.88/1.82	      4: l0 -> l1 : A'=B, B'=0, [ B>=1 ], cost: 1+B
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	### Computing asymptotic complexity ###
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	Fully simplified ITS problem
3.88/1.82	
3.88/1.82	   Start location: l0
3.88/1.82	
3.88/1.82	      4: l0 -> l1 : A'=B, B'=0, [ B>=1 ], cost: 1+B
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	Computing asymptotic complexity for rule 4
3.88/1.82	
3.88/1.82	   Solved the limit problem by the following transformations:
3.88/1.82	
3.88/1.82	      Created initial limit problem:
3.88/1.82	
3.88/1.82	      1+B (+), B (+/+!) [not solved]
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	      removing all constraints (solved by SMT)
3.88/1.82	
3.88/1.82	      resulting limit problem: [solved]
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	      applying transformation rule (C) using substitution {B==n}
3.88/1.82	
3.88/1.82	      resulting limit problem:
3.88/1.82	
3.88/1.82	      [solved]
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	   Solution:
3.88/1.82	
3.88/1.82	      B / n
3.88/1.82	
3.88/1.82	   Resulting cost 1+n has complexity: Poly(n^1)
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	Found new complexity Poly(n^1).
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	Obtained the following overall complexity (w.r.t. the length of the input n):
3.88/1.82	
3.88/1.82	   Complexity:  Poly(n^1)
3.88/1.82	
3.88/1.82	   Cpx degree:  1
3.88/1.82	
3.88/1.82	   Solved cost: 1+n
3.88/1.82	
3.88/1.82	   Rule cost:   1+B
3.88/1.82	
3.88/1.82	   Rule guard:  [ B>=1 ]
3.88/1.82	
3.88/1.82	
3.88/1.82	
3.88/1.82	WORST_CASE(Omega(n^1),?)
3.88/1.82	
3.88/1.82	
3.88/1.82	----------------------------------------
3.88/1.82	
3.88/1.82	(4)
3.88/1.82	BOUNDS(n^1, INF)
3.88/1.83	EOF