3.74/1.87	WORST_CASE(Omega(n^1), O(n^1))
3.74/1.88	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.74/1.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.74/1.88	
3.74/1.88	
3.74/1.88	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
3.74/1.88	
3.74/1.88	(0) CpxIntTrs
3.74/1.88	(1) Koat Proof [FINISHED, 26 ms]
3.74/1.88	(2) BOUNDS(1, n^1)
3.74/1.88	(3) Loat Proof [FINISHED, 220 ms]
3.74/1.88	(4) BOUNDS(n^1, INF)
3.74/1.88	
3.74/1.88	
3.74/1.88	----------------------------------------
3.74/1.88	
3.74/1.88	(0)
3.74/1.88	Obligation:
3.74/1.88	Complexity Int TRS consisting of the following rules:
3.74/1.88	f300(A, B, C) -> Com_1(f300(-(1) + A, -(2) + A, C)) :|: A >= 1 && B + A >= 1 && B >= 1
3.74/1.88	f300(A, B, C) -> Com_1(f1(A, B, D)) :|: A >= 1 && 0 >= B + A && B >= 1
3.74/1.88	f300(A, B, C) -> Com_1(f1(A, B, D)) :|: B >= 1 && 0 >= A
3.74/1.88	f300(A, B, C) -> Com_1(f1(A, B, D)) :|: 0 >= B
3.74/1.88	f2(A, B, C) -> Com_1(f300(A, B, C)) :|: TRUE
3.74/1.88	
3.74/1.88	The start-symbols are:[f2_3]
3.74/1.88	
3.74/1.88	
3.74/1.88	----------------------------------------
3.74/1.88	
3.74/1.88	(1) Koat Proof (FINISHED)
3.74/1.88	YES(?, ar_0 + 3)
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Initial complexity problem:
3.74/1.88	
3.74/1.88	1:	T:
3.74/1.88	
3.74/1.88			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f300(ar_0 - 1, ar_0 - 2, ar_2)) [ ar_0 >= 1 /\ ar_1 + ar_0 >= 1 /\ ar_1 >= 1 ]
3.74/1.88	
3.74/1.88			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, d)) [ ar_0 >= 1 /\ 0 >= ar_1 + ar_0 /\ ar_1 >= 1 ]
3.74/1.88	
3.74/1.88			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, d)) [ ar_1 >= 1 /\ 0 >= ar_0 ]
3.74/1.88	
3.74/1.88			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, d)) [ 0 >= ar_1 ]
3.74/1.88	
3.74/1.88			(Comp: ?, Cost: 1)    f2(ar_0, ar_1, ar_2) -> Com_1(f300(ar_0, ar_1, ar_2))
3.74/1.88	
3.74/1.88			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f2(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.74/1.88	
3.74/1.88		start location:	koat_start
3.74/1.88	
3.74/1.88		leaf cost:	0
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Testing for reachability in the complexity graph removes the following transition from problem 1:
3.74/1.88	
3.74/1.88		f300(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, d)) [ ar_0 >= 1 /\ 0 >= ar_1 + ar_0 /\ ar_1 >= 1 ]
3.74/1.88	
3.74/1.88	We thus obtain the following problem:
3.74/1.88	
3.74/1.88	2:	T:
3.74/1.88	
3.74/1.88			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, d)) [ 0 >= ar_1 ]
3.74/1.88	
3.74/1.88			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, d)) [ ar_1 >= 1 /\ 0 >= ar_0 ]
3.74/1.88	
3.74/1.88			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f300(ar_0 - 1, ar_0 - 2, ar_2)) [ ar_0 >= 1 /\ ar_1 + ar_0 >= 1 /\ ar_1 >= 1 ]
3.74/1.88	
3.74/1.88			(Comp: ?, Cost: 1)    f2(ar_0, ar_1, ar_2) -> Com_1(f300(ar_0, ar_1, ar_2))
3.74/1.88	
3.74/1.88			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f2(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.74/1.88	
3.74/1.88		start location:	koat_start
3.74/1.88	
3.74/1.88		leaf cost:	0
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Repeatedly propagating knowledge in problem 2 produces the following problem:
3.74/1.88	
3.74/1.88	3:	T:
3.74/1.88	
3.74/1.88			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, d)) [ 0 >= ar_1 ]
3.74/1.88	
3.74/1.88			(Comp: 1, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, d)) [ ar_1 >= 1 /\ 0 >= ar_0 ]
3.74/1.88	
3.74/1.88			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f300(ar_0 - 1, ar_0 - 2, ar_2)) [ ar_0 >= 1 /\ ar_1 + ar_0 >= 1 /\ ar_1 >= 1 ]
3.74/1.88	
3.74/1.88			(Comp: 1, Cost: 1)    f2(ar_0, ar_1, ar_2) -> Com_1(f300(ar_0, ar_1, ar_2))
3.74/1.88	
3.74/1.88			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f2(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.74/1.88	
3.74/1.88		start location:	koat_start
3.74/1.88	
3.74/1.88		leaf cost:	0
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	A polynomial rank function with
3.74/1.88	
3.74/1.88		Pol(f300) = 1
3.74/1.88	
3.74/1.88		Pol(f1) = 0
3.74/1.88	
3.74/1.88		Pol(f2) = 1
3.74/1.88	
3.74/1.88		Pol(koat_start) = 1
3.74/1.88	
3.74/1.88	orients all transitions weakly and the transition
3.74/1.88	
3.74/1.88		f300(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, d)) [ 0 >= ar_1 ]
3.74/1.88	
3.74/1.88	strictly and produces the following problem:
3.74/1.88	
3.74/1.88	4:	T:
3.74/1.88	
3.74/1.88			(Comp: 1, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, d)) [ 0 >= ar_1 ]
3.74/1.88	
3.74/1.88			(Comp: 1, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, d)) [ ar_1 >= 1 /\ 0 >= ar_0 ]
3.74/1.88	
3.74/1.88			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f300(ar_0 - 1, ar_0 - 2, ar_2)) [ ar_0 >= 1 /\ ar_1 + ar_0 >= 1 /\ ar_1 >= 1 ]
3.74/1.88	
3.74/1.88			(Comp: 1, Cost: 1)    f2(ar_0, ar_1, ar_2) -> Com_1(f300(ar_0, ar_1, ar_2))
3.74/1.88	
3.74/1.88			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f2(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.74/1.88	
3.74/1.88		start location:	koat_start
3.74/1.88	
3.74/1.88		leaf cost:	0
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	A polynomial rank function with
3.74/1.88	
3.74/1.88		Pol(f300) = V_1
3.74/1.88	
3.74/1.88		Pol(f1) = V_1
3.74/1.88	
3.74/1.88		Pol(f2) = V_1
3.74/1.88	
3.74/1.88		Pol(koat_start) = V_1
3.74/1.88	
3.74/1.88	orients all transitions weakly and the transition
3.74/1.88	
3.74/1.88		f300(ar_0, ar_1, ar_2) -> Com_1(f300(ar_0 - 1, ar_0 - 2, ar_2)) [ ar_0 >= 1 /\ ar_1 + ar_0 >= 1 /\ ar_1 >= 1 ]
3.74/1.88	
3.74/1.88	strictly and produces the following problem:
3.74/1.88	
3.74/1.88	5:	T:
3.74/1.88	
3.74/1.88			(Comp: 1, Cost: 1)       f300(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, d)) [ 0 >= ar_1 ]
3.74/1.88	
3.74/1.88			(Comp: 1, Cost: 1)       f300(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, d)) [ ar_1 >= 1 /\ 0 >= ar_0 ]
3.74/1.88	
3.74/1.88			(Comp: ar_0, Cost: 1)    f300(ar_0, ar_1, ar_2) -> Com_1(f300(ar_0 - 1, ar_0 - 2, ar_2)) [ ar_0 >= 1 /\ ar_1 + ar_0 >= 1 /\ ar_1 >= 1 ]
3.74/1.88	
3.74/1.88			(Comp: 1, Cost: 1)       f2(ar_0, ar_1, ar_2) -> Com_1(f300(ar_0, ar_1, ar_2))
3.74/1.88	
3.74/1.88			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1, ar_2) -> Com_1(f2(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.74/1.88	
3.74/1.88		start location:	koat_start
3.74/1.88	
3.74/1.88		leaf cost:	0
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Complexity upper bound ar_0 + 3
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Time: 0.071 sec (SMT: 0.066 sec)
3.74/1.88	
3.74/1.88	
3.74/1.88	----------------------------------------
3.74/1.88	
3.74/1.88	(2)
3.74/1.88	BOUNDS(1, n^1)
3.74/1.88	
3.74/1.88	----------------------------------------
3.74/1.88	
3.74/1.88	(3) Loat Proof (FINISHED)
3.74/1.88	
3.74/1.88	
3.74/1.88	### Pre-processing the ITS problem ###
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Initial linear ITS problem
3.74/1.88	
3.74/1.88	   Start location: f2
3.74/1.88	
3.74/1.88	      0: f300 -> f300 : A'=-1+A, B'=-2+A, [ A>=1 && A+B>=1 && B>=1 ], cost: 1
3.74/1.88	
3.74/1.88	      1: f300 -> f1 : C'=free, [ A>=1 && 0>=A+B && B>=1 ], cost: 1
3.74/1.88	
3.74/1.88	      2: f300 -> f1 : C'=free_1, [ B>=1 && 0>=A ], cost: 1
3.74/1.88	
3.74/1.88	      3: f300 -> f1 : C'=free_2, [ 0>=B ], cost: 1
3.74/1.88	
3.74/1.88	      4: f2 -> f300 : [], cost: 1
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Removed unreachable and leaf rules:
3.74/1.88	
3.74/1.88	   Start location: f2
3.74/1.88	
3.74/1.88	      0: f300 -> f300 : A'=-1+A, B'=-2+A, [ A>=1 && A+B>=1 && B>=1 ], cost: 1
3.74/1.88	
3.74/1.88	      4: f2 -> f300 : [], cost: 1
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	### Simplification by acceleration and chaining ###
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Accelerating simple loops of location 0.
3.74/1.88	
3.74/1.88	   Accelerating the following rules:
3.74/1.88	
3.74/1.88	      0: f300 -> f300 : A'=-1+A, B'=-2+A, [ A>=1 && A+B>=1 && B>=1 ], cost: 1
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	   Accelerated rule 0 with backward acceleration, yielding the new rule 5.
3.74/1.88	
3.74/1.88	   Accelerated rule 0 with backward acceleration, yielding the new rule 6.
3.74/1.88	
3.74/1.88	   Accelerated rule 0 with backward acceleration, yielding the new rule 7.
3.74/1.88	
3.74/1.88	   Removing the simple loops: 0.
3.74/1.88	
3.74/1.88	   Also removing duplicate rules:.
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Accelerated all simple loops using metering functions (where possible):
3.74/1.88	
3.74/1.88	   Start location: f2
3.74/1.88	
3.74/1.88	      6: f300 -> f300 : A'=0, B'=-1, [ 0>=1 ], cost: A
3.74/1.88	
3.74/1.88	      7: f300 -> f300 : A'=1, B'=0, [ A+B>=1 && B>=1 && -1+A>0 ], cost: -1+A
3.74/1.88	
3.74/1.88	      4: f2 -> f300 : [], cost: 1
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Chained accelerated rules (with incoming rules):
3.74/1.88	
3.74/1.88	   Start location: f2
3.74/1.88	
3.74/1.88	      4: f2 -> f300 : [], cost: 1
3.74/1.88	
3.74/1.88	      8: f2 -> f300 : A'=1, B'=0, [ A+B>=1 && B>=1 && -1+A>0 ], cost: A
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Removed unreachable locations (and leaf rules with constant cost):
3.74/1.88	
3.74/1.88	   Start location: f2
3.74/1.88	
3.74/1.88	      8: f2 -> f300 : A'=1, B'=0, [ A+B>=1 && B>=1 && -1+A>0 ], cost: A
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	### Computing asymptotic complexity ###
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Fully simplified ITS problem
3.74/1.88	
3.74/1.88	   Start location: f2
3.74/1.88	
3.74/1.88	      8: f2 -> f300 : A'=1, B'=0, [ A+B>=1 && B>=1 && -1+A>0 ], cost: A
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Computing asymptotic complexity for rule 8
3.74/1.88	
3.74/1.88	   Simplified the guard:
3.74/1.88	
3.74/1.88	      8: f2 -> f300 : A'=1, B'=0, [ B>=1 && -1+A>0 ], cost: A
3.74/1.88	
3.74/1.88	   Solved the limit problem by the following transformations:
3.74/1.88	
3.74/1.88	      Created initial limit problem:
3.74/1.88	
3.74/1.88	      -1+A (+/+!), A (+), B (+/+!) [not solved]
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	      applying transformation rule (C) using substitution {B==1}
3.74/1.88	
3.74/1.88	      resulting limit problem:
3.74/1.88	
3.74/1.88	      1 (+/+!), -1+A (+/+!), A (+) [not solved]
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	      applying transformation rule (B), deleting 1 (+/+!)
3.74/1.88	
3.74/1.88	      resulting limit problem:
3.74/1.88	
3.74/1.88	      -1+A (+/+!), A (+) [not solved]
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	      removing all constraints (solved by SMT)
3.74/1.88	
3.74/1.88	      resulting limit problem: [solved]
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	      applying transformation rule (C) using substitution {A==n}
3.74/1.88	
3.74/1.88	      resulting limit problem:
3.74/1.88	
3.74/1.88	      [solved]
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	   Solution:
3.74/1.88	
3.74/1.88	      A / n
3.74/1.88	
3.74/1.88	      B / 1
3.74/1.88	
3.74/1.88	   Resulting cost n has complexity: Poly(n^1)
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Found new complexity Poly(n^1).
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	Obtained the following overall complexity (w.r.t. the length of the input n):
3.74/1.88	
3.74/1.88	   Complexity:  Poly(n^1)
3.74/1.88	
3.74/1.88	   Cpx degree:  1
3.74/1.88	
3.74/1.88	   Solved cost: n
3.74/1.88	
3.74/1.88	   Rule cost:   A
3.74/1.88	
3.74/1.88	   Rule guard:  [ B>=1 && -1+A>0 ]
3.74/1.88	
3.74/1.88	
3.74/1.88	
3.74/1.88	WORST_CASE(Omega(n^1),?)
3.74/1.88	
3.74/1.88	
3.74/1.88	----------------------------------------
3.74/1.88	
3.74/1.88	(4)
3.74/1.88	BOUNDS(n^1, INF)
3.93/1.90	EOF