3.97/1.93	WORST_CASE(Omega(n^1), O(n^1))
3.97/1.94	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
3.97/1.94	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.97/1.94	
3.97/1.94	
3.97/1.94	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
3.97/1.94	
3.97/1.94	(0) CpxIntTrs
3.97/1.94	(1) Koat Proof [FINISHED, 132 ms]
3.97/1.94	(2) BOUNDS(1, n^1)
3.97/1.94	(3) Loat Proof [FINISHED, 198 ms]
3.97/1.94	(4) BOUNDS(n^1, INF)
3.97/1.94	
3.97/1.94	
3.97/1.94	----------------------------------------
3.97/1.94	
3.97/1.94	(0)
3.97/1.94	Obligation:
3.97/1.94	Complexity Int TRS consisting of the following rules:
3.97/1.94	f2(A, B, C, D, E, F) -> Com_1(f2(-(1) + A, -(1) + B, A, B, -(2) + A, F)) :|: A >= 1 && B >= 1
3.97/1.94	f3(A, B, C, D, E, F) -> Com_1(f2(A, B, C, D, E, F)) :|: TRUE
3.97/1.94	f2(A, B, C, D, E, F) -> Com_1(f4(A, G, C, D, E, H)) :|: 0 >= B && 0 >= G
3.97/1.94	f2(A, B, C, D, E, F) -> Com_1(f4(A, B, C, D, E, H)) :|: B >= 1 && 0 >= A
3.97/1.94	
3.97/1.94	The start-symbols are:[f3_6]
3.97/1.94	
3.97/1.94	
3.97/1.94	----------------------------------------
3.97/1.94	
3.97/1.94	(1) Koat Proof (FINISHED)
3.97/1.94	YES(?, ar_0 + 3)
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Initial complexity problem:
3.97/1.94	
3.97/1.94	1:	T:
3.97/1.94	
3.97/1.94			(Comp: ?, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f2(ar_0 - 1, ar_1 - 1, ar_0, ar_1, ar_0 - 2, ar_5)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
3.97/1.94	
3.97/1.94			(Comp: ?, Cost: 1)    f3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
3.97/1.94	
3.97/1.94			(Comp: ?, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f4(ar_0, g, ar_2, ar_3, ar_4, h)) [ 0 >= ar_1 /\ 0 >= g ]
3.97/1.94	
3.97/1.94			(Comp: ?, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f4(ar_0, ar_1, ar_2, ar_3, ar_4, g)) [ ar_1 >= 1 /\ 0 >= ar_0 ]
3.97/1.94	
3.97/1.94			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
3.97/1.94	
3.97/1.94		start location:	koat_start
3.97/1.94	
3.97/1.94		leaf cost:	0
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.97/1.94	
3.97/1.94	2:	T:
3.97/1.94	
3.97/1.94			(Comp: ?, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f2(ar_0 - 1, ar_1 - 1, ar_0, ar_1, ar_0 - 2, ar_5)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
3.97/1.94	
3.97/1.94			(Comp: 1, Cost: 1)    f3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
3.97/1.94	
3.97/1.94			(Comp: ?, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f4(ar_0, g, ar_2, ar_3, ar_4, h)) [ 0 >= ar_1 /\ 0 >= g ]
3.97/1.94	
3.97/1.94			(Comp: ?, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f4(ar_0, ar_1, ar_2, ar_3, ar_4, g)) [ ar_1 >= 1 /\ 0 >= ar_0 ]
3.97/1.94	
3.97/1.94			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
3.97/1.94	
3.97/1.94		start location:	koat_start
3.97/1.94	
3.97/1.94		leaf cost:	0
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	A polynomial rank function with
3.97/1.94	
3.97/1.94		Pol(f2) = 1
3.97/1.94	
3.97/1.94		Pol(f3) = 1
3.97/1.94	
3.97/1.94		Pol(f4) = 0
3.97/1.94	
3.97/1.94		Pol(koat_start) = 1
3.97/1.94	
3.97/1.94	orients all transitions weakly and the transitions
3.97/1.94	
3.97/1.94		f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f4(ar_0, g, ar_2, ar_3, ar_4, h)) [ 0 >= ar_1 /\ 0 >= g ]
3.97/1.94	
3.97/1.94		f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f4(ar_0, ar_1, ar_2, ar_3, ar_4, g)) [ ar_1 >= 1 /\ 0 >= ar_0 ]
3.97/1.94	
3.97/1.94	strictly and produces the following problem:
3.97/1.94	
3.97/1.94	3:	T:
3.97/1.94	
3.97/1.94			(Comp: ?, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f2(ar_0 - 1, ar_1 - 1, ar_0, ar_1, ar_0 - 2, ar_5)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
3.97/1.94	
3.97/1.94			(Comp: 1, Cost: 1)    f3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
3.97/1.94	
3.97/1.94			(Comp: 1, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f4(ar_0, g, ar_2, ar_3, ar_4, h)) [ 0 >= ar_1 /\ 0 >= g ]
3.97/1.94	
3.97/1.94			(Comp: 1, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f4(ar_0, ar_1, ar_2, ar_3, ar_4, g)) [ ar_1 >= 1 /\ 0 >= ar_0 ]
3.97/1.94	
3.97/1.94			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
3.97/1.94	
3.97/1.94		start location:	koat_start
3.97/1.94	
3.97/1.94		leaf cost:	0
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	A polynomial rank function with
3.97/1.94	
3.97/1.94		Pol(f2) = V_1
3.97/1.94	
3.97/1.94		Pol(f3) = V_1
3.97/1.94	
3.97/1.94		Pol(f4) = V_1
3.97/1.94	
3.97/1.94		Pol(koat_start) = V_1
3.97/1.94	
3.97/1.94	orients all transitions weakly and the transition
3.97/1.94	
3.97/1.94		f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f2(ar_0 - 1, ar_1 - 1, ar_0, ar_1, ar_0 - 2, ar_5)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
3.97/1.94	
3.97/1.94	strictly and produces the following problem:
3.97/1.94	
3.97/1.94	4:	T:
3.97/1.94	
3.97/1.94			(Comp: ar_0, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f2(ar_0 - 1, ar_1 - 1, ar_0, ar_1, ar_0 - 2, ar_5)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
3.97/1.94	
3.97/1.94			(Comp: 1, Cost: 1)       f3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
3.97/1.94	
3.97/1.94			(Comp: 1, Cost: 1)       f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f4(ar_0, g, ar_2, ar_3, ar_4, h)) [ 0 >= ar_1 /\ 0 >= g ]
3.97/1.94	
3.97/1.94			(Comp: 1, Cost: 1)       f2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f4(ar_0, ar_1, ar_2, ar_3, ar_4, g)) [ ar_1 >= 1 /\ 0 >= ar_0 ]
3.97/1.94	
3.97/1.94			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
3.97/1.94	
3.97/1.94		start location:	koat_start
3.97/1.94	
3.97/1.94		leaf cost:	0
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Complexity upper bound ar_0 + 3
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Time: 0.111 sec (SMT: 0.103 sec)
3.97/1.94	
3.97/1.94	
3.97/1.94	----------------------------------------
3.97/1.94	
3.97/1.94	(2)
3.97/1.94	BOUNDS(1, n^1)
3.97/1.94	
3.97/1.94	----------------------------------------
3.97/1.94	
3.97/1.94	(3) Loat Proof (FINISHED)
3.97/1.94	
3.97/1.94	
3.97/1.94	### Pre-processing the ITS problem ###
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Initial linear ITS problem
3.97/1.94	
3.97/1.94	   Start location: f3
3.97/1.94	
3.97/1.94	      0: f2 -> f2 : A'=-1+A, B'=-1+B, C'=A, D'=B, E'=-2+A, [ A>=1 && B>=1 ], cost: 1
3.97/1.94	
3.97/1.94	      2: f2 -> f4 : B'=free, F'=free_1, [ 0>=B && 0>=free ], cost: 1
3.97/1.94	
3.97/1.94	      3: f2 -> f4 : F'=free_2, [ B>=1 && 0>=A ], cost: 1
3.97/1.94	
3.97/1.94	      1: f3 -> f2 : [], cost: 1
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Removed unreachable and leaf rules:
3.97/1.94	
3.97/1.94	   Start location: f3
3.97/1.94	
3.97/1.94	      0: f2 -> f2 : A'=-1+A, B'=-1+B, C'=A, D'=B, E'=-2+A, [ A>=1 && B>=1 ], cost: 1
3.97/1.94	
3.97/1.94	      1: f3 -> f2 : [], cost: 1
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	### Simplification by acceleration and chaining ###
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Accelerating simple loops of location 0.
3.97/1.94	
3.97/1.94	   Accelerating the following rules:
3.97/1.94	
3.97/1.94	      0: f2 -> f2 : A'=-1+A, B'=-1+B, C'=A, D'=B, E'=-2+A, [ A>=1 && B>=1 ], cost: 1
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	   Accelerated rule 0 with metering function B (after adding A>=B), yielding the new rule 4.
3.97/1.94	
3.97/1.94	   Accelerated rule 0 with metering function A (after adding A<=B), yielding the new rule 5.
3.97/1.94	
3.97/1.94	   Removing the simple loops: 0.
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Accelerated all simple loops using metering functions (where possible):
3.97/1.94	
3.97/1.94	   Start location: f3
3.97/1.94	
3.97/1.94	      4: f2 -> f2 : A'=A-B, B'=0, C'=1+A-B, D'=1, E'=-1+A-B, [ A>=1 && B>=1 && A>=B ], cost: B
3.97/1.94	
3.97/1.94	      5: f2 -> f2 : A'=0, B'=-A+B, C'=1, D'=1-A+B, E'=-1, [ A>=1 && B>=1 && A<=B ], cost: A
3.97/1.94	
3.97/1.94	      1: f3 -> f2 : [], cost: 1
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Chained accelerated rules (with incoming rules):
3.97/1.94	
3.97/1.94	   Start location: f3
3.97/1.94	
3.97/1.94	      1: f3 -> f2 : [], cost: 1
3.97/1.94	
3.97/1.94	      6: f3 -> f2 : A'=A-B, B'=0, C'=1+A-B, D'=1, E'=-1+A-B, [ A>=1 && B>=1 && A>=B ], cost: 1+B
3.97/1.94	
3.97/1.94	      7: f3 -> f2 : A'=0, B'=-A+B, C'=1, D'=1-A+B, E'=-1, [ A>=1 && B>=1 && A<=B ], cost: 1+A
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Removed unreachable locations (and leaf rules with constant cost):
3.97/1.94	
3.97/1.94	   Start location: f3
3.97/1.94	
3.97/1.94	      6: f3 -> f2 : A'=A-B, B'=0, C'=1+A-B, D'=1, E'=-1+A-B, [ A>=1 && B>=1 && A>=B ], cost: 1+B
3.97/1.94	
3.97/1.94	      7: f3 -> f2 : A'=0, B'=-A+B, C'=1, D'=1-A+B, E'=-1, [ A>=1 && B>=1 && A<=B ], cost: 1+A
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	### Computing asymptotic complexity ###
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Fully simplified ITS problem
3.97/1.94	
3.97/1.94	   Start location: f3
3.97/1.94	
3.97/1.94	      6: f3 -> f2 : A'=A-B, B'=0, C'=1+A-B, D'=1, E'=-1+A-B, [ A>=1 && B>=1 && A>=B ], cost: 1+B
3.97/1.94	
3.97/1.94	      7: f3 -> f2 : A'=0, B'=-A+B, C'=1, D'=1-A+B, E'=-1, [ A>=1 && B>=1 && A<=B ], cost: 1+A
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Computing asymptotic complexity for rule 6
3.97/1.94	
3.97/1.94	   Simplified the guard:
3.97/1.94	
3.97/1.94	      6: f3 -> f2 : A'=A-B, B'=0, C'=1+A-B, D'=1, E'=-1+A-B, [ B>=1 && A>=B ], cost: 1+B
3.97/1.94	
3.97/1.94	   Solved the limit problem by the following transformations:
3.97/1.94	
3.97/1.94	      Created initial limit problem:
3.97/1.94	
3.97/1.94	      1+B (+), 1+A-B (+/+!), B (+/+!) [not solved]
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	      removing all constraints (solved by SMT)
3.97/1.94	
3.97/1.94	      resulting limit problem: [solved]
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	      applying transformation rule (C) using substitution {A==n,B==n}
3.97/1.94	
3.97/1.94	      resulting limit problem:
3.97/1.94	
3.97/1.94	      [solved]
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	   Solution:
3.97/1.94	
3.97/1.94	      A / n
3.97/1.94	
3.97/1.94	      B / n
3.97/1.94	
3.97/1.94	   Resulting cost 1+n has complexity: Poly(n^1)
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Found new complexity Poly(n^1).
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	Obtained the following overall complexity (w.r.t. the length of the input n):
3.97/1.94	
3.97/1.94	   Complexity:  Poly(n^1)
3.97/1.94	
3.97/1.94	   Cpx degree:  1
3.97/1.94	
3.97/1.94	   Solved cost: 1+n
3.97/1.94	
3.97/1.94	   Rule cost:   1+B
3.97/1.94	
3.97/1.94	   Rule guard:  [ B>=1 && A>=B ]
3.97/1.94	
3.97/1.94	
3.97/1.94	
3.97/1.94	WORST_CASE(Omega(n^1),?)
3.97/1.94	
3.97/1.94	
3.97/1.94	----------------------------------------
3.97/1.94	
3.97/1.94	(4)
3.97/1.94	BOUNDS(n^1, INF)
3.97/1.96	EOF