4.82/2.34	WORST_CASE(Omega(n^1), O(n^1))
4.82/2.35	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.82/2.35	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.82/2.35	
4.82/2.35	
4.82/2.35	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
4.82/2.35	
4.82/2.35	(0) CpxIntTrs
4.82/2.35	(1) Koat Proof [FINISHED, 39 ms]
4.82/2.35	(2) BOUNDS(1, n^1)
4.82/2.35	(3) Loat Proof [FINISHED, 736 ms]
4.82/2.35	(4) BOUNDS(n^1, INF)
4.82/2.35	
4.82/2.35	
4.82/2.35	----------------------------------------
4.82/2.35	
4.82/2.35	(0)
4.82/2.35	Obligation:
4.82/2.35	Complexity Int TRS consisting of the following rules:
4.82/2.35	f2(A, B, C, D) -> Com_1(f300(A, B, C, D)) :|: TRUE
4.82/2.35	f300(A, B, C, D) -> Com_1(f300(1 + A, B, E, D)) :|: E >= 1 && B >= 1 + A
4.82/2.35	f300(A, B, C, D) -> Com_1(f300(1 + A, B, E, D)) :|: 0 >= E + 1 && B >= 1 + A
4.82/2.35	f300(A, B, C, D) -> Com_1(f300(A, -(1) + B, 0, D)) :|: B >= 1 + A
4.82/2.35	f300(A, B, C, D) -> Com_1(f1(A, B, C, E)) :|: A >= B
4.82/2.35	
4.82/2.35	The start-symbols are:[f2_4]
4.82/2.35	
4.82/2.35	
4.82/2.35	----------------------------------------
4.82/2.35	
4.82/2.35	(1) Koat Proof (FINISHED)
4.82/2.35	YES(?, 3*ar_0 + 3*ar_1 + 2)
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Initial complexity problem:
4.82/2.35	
4.82/2.35	1:	T:
4.82/2.35	
4.82/2.35			(Comp: ?, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3))
4.82/2.35	
4.82/2.35			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1 + 1, e, ar_3)) [ e >= 1 /\ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1 + 1, e, ar_3)) [ 0 >= e + 1 /\ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0 - 1, ar_1, 0, ar_3)) [ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, ar_2, e)) [ ar_1 >= ar_0 ]
4.82/2.35	
4.82/2.35			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f2(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.82/2.35	
4.82/2.35		start location:	koat_start
4.82/2.35	
4.82/2.35		leaf cost:	0
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Repeatedly propagating knowledge in problem 1 produces the following problem:
4.82/2.35	
4.82/2.35	2:	T:
4.82/2.35	
4.82/2.35			(Comp: 1, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3))
4.82/2.35	
4.82/2.35			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1 + 1, e, ar_3)) [ e >= 1 /\ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1 + 1, e, ar_3)) [ 0 >= e + 1 /\ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0 - 1, ar_1, 0, ar_3)) [ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, ar_2, e)) [ ar_1 >= ar_0 ]
4.82/2.35	
4.82/2.35			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f2(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.82/2.35	
4.82/2.35		start location:	koat_start
4.82/2.35	
4.82/2.35		leaf cost:	0
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	A polynomial rank function with
4.82/2.35	
4.82/2.35		Pol(f2) = 1
4.82/2.35	
4.82/2.35		Pol(f300) = 1
4.82/2.35	
4.82/2.35		Pol(f1) = 0
4.82/2.35	
4.82/2.35		Pol(koat_start) = 1
4.82/2.35	
4.82/2.35	orients all transitions weakly and the transition
4.82/2.35	
4.82/2.35		f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, ar_2, e)) [ ar_1 >= ar_0 ]
4.82/2.35	
4.82/2.35	strictly and produces the following problem:
4.82/2.35	
4.82/2.35	3:	T:
4.82/2.35	
4.82/2.35			(Comp: 1, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3))
4.82/2.35	
4.82/2.35			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1 + 1, e, ar_3)) [ e >= 1 /\ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1 + 1, e, ar_3)) [ 0 >= e + 1 /\ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35			(Comp: ?, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0 - 1, ar_1, 0, ar_3)) [ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35			(Comp: 1, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, ar_2, e)) [ ar_1 >= ar_0 ]
4.82/2.35	
4.82/2.35			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f2(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.82/2.35	
4.82/2.35		start location:	koat_start
4.82/2.35	
4.82/2.35		leaf cost:	0
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	A polynomial rank function with
4.82/2.35	
4.82/2.35		Pol(f2) = V_1 - V_2
4.82/2.35	
4.82/2.35		Pol(f300) = V_1 - V_2
4.82/2.35	
4.82/2.35		Pol(f1) = V_1 - V_2
4.82/2.35	
4.82/2.35		Pol(koat_start) = V_1 - V_2
4.82/2.35	
4.82/2.35	orients all transitions weakly and the transitions
4.82/2.35	
4.82/2.35		f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1 + 1, e, ar_3)) [ e >= 1 /\ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35		f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1 + 1, e, ar_3)) [ 0 >= e + 1 /\ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35		f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0 - 1, ar_1, 0, ar_3)) [ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35	strictly and produces the following problem:
4.82/2.35	
4.82/2.35	4:	T:
4.82/2.35	
4.82/2.35			(Comp: 1, Cost: 1)              f2(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1, ar_2, ar_3))
4.82/2.35	
4.82/2.35			(Comp: ar_0 + ar_1, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1 + 1, e, ar_3)) [ e >= 1 /\ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35			(Comp: ar_0 + ar_1, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0, ar_1 + 1, e, ar_3)) [ 0 >= e + 1 /\ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35			(Comp: ar_0 + ar_1, Cost: 1)    f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f300(ar_0 - 1, ar_1, 0, ar_3)) [ ar_0 >= ar_1 + 1 ]
4.82/2.35	
4.82/2.35			(Comp: 1, Cost: 1)              f300(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, ar_2, e)) [ ar_1 >= ar_0 ]
4.82/2.35	
4.82/2.35			(Comp: 1, Cost: 0)              koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f2(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.82/2.35	
4.82/2.35		start location:	koat_start
4.82/2.35	
4.82/2.35		leaf cost:	0
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Complexity upper bound 3*ar_0 + 3*ar_1 + 2
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Time: 0.103 sec (SMT: 0.094 sec)
4.82/2.35	
4.82/2.35	
4.82/2.35	----------------------------------------
4.82/2.35	
4.82/2.35	(2)
4.82/2.35	BOUNDS(1, n^1)
4.82/2.35	
4.82/2.35	----------------------------------------
4.82/2.35	
4.82/2.35	(3) Loat Proof (FINISHED)
4.82/2.35	
4.82/2.35	
4.82/2.35	### Pre-processing the ITS problem ###
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Initial linear ITS problem
4.82/2.35	
4.82/2.35	   Start location: f2
4.82/2.35	
4.82/2.35	      0: f2 -> f300 : [], cost: 1
4.82/2.35	
4.82/2.35	      1: f300 -> f300 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1
4.82/2.35	
4.82/2.35	      2: f300 -> f300 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1
4.82/2.35	
4.82/2.35	      3: f300 -> f300 : A'=-1+A, C'=0, [ A>=1+B ], cost: 1
4.82/2.35	
4.82/2.35	      4: f300 -> f1 : D'=free_2, [ B>=A ], cost: 1
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Removed unreachable and leaf rules:
4.82/2.35	
4.82/2.35	   Start location: f2
4.82/2.35	
4.82/2.35	      0: f2 -> f300 : [], cost: 1
4.82/2.35	
4.82/2.35	      1: f300 -> f300 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1
4.82/2.35	
4.82/2.35	      2: f300 -> f300 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1
4.82/2.35	
4.82/2.35	      3: f300 -> f300 : A'=-1+A, C'=0, [ A>=1+B ], cost: 1
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	### Simplification by acceleration and chaining ###
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Accelerating simple loops of location 1.
4.82/2.35	
4.82/2.35	   Accelerating the following rules:
4.82/2.35	
4.82/2.35	      1: f300 -> f300 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1
4.82/2.35	
4.82/2.35	      2: f300 -> f300 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1
4.82/2.35	
4.82/2.35	      3: f300 -> f300 : A'=-1+A, C'=0, [ A>=1+B ], cost: 1
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	   Accelerated rule 1 with metering function A-B, yielding the new rule 5.
4.82/2.35	
4.82/2.35	   Accelerated rule 2 with metering function A-B, yielding the new rule 6.
4.82/2.35	
4.82/2.35	   Accelerated rule 3 with metering function A-B, yielding the new rule 7.
4.82/2.35	
4.82/2.35	   Removing the simple loops: 1 2 3.
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Accelerated all simple loops using metering functions (where possible):
4.82/2.35	
4.82/2.35	   Start location: f2
4.82/2.35	
4.82/2.35	      0: f2 -> f300 : [], cost: 1
4.82/2.35	
4.82/2.35	      5: f300 -> f300 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: A-B
4.82/2.35	
4.82/2.35	      6: f300 -> f300 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: A-B
4.82/2.35	
4.82/2.35	      7: f300 -> f300 : A'=B, C'=0, [ A>=1+B ], cost: A-B
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Chained accelerated rules (with incoming rules):
4.82/2.35	
4.82/2.35	   Start location: f2
4.82/2.35	
4.82/2.35	      0: f2 -> f300 : [], cost: 1
4.82/2.35	
4.82/2.35	      8: f2 -> f300 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: 1+A-B
4.82/2.35	
4.82/2.35	      9: f2 -> f300 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1+A-B
4.82/2.35	
4.82/2.35	     10: f2 -> f300 : A'=B, C'=0, [ A>=1+B ], cost: 1+A-B
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Removed unreachable locations (and leaf rules with constant cost):
4.82/2.35	
4.82/2.35	   Start location: f2
4.82/2.35	
4.82/2.35	      8: f2 -> f300 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: 1+A-B
4.82/2.35	
4.82/2.35	      9: f2 -> f300 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1+A-B
4.82/2.35	
4.82/2.35	     10: f2 -> f300 : A'=B, C'=0, [ A>=1+B ], cost: 1+A-B
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	### Computing asymptotic complexity ###
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Fully simplified ITS problem
4.82/2.35	
4.82/2.35	   Start location: f2
4.82/2.35	
4.82/2.35	      8: f2 -> f300 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: 1+A-B
4.82/2.35	
4.82/2.35	      9: f2 -> f300 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1+A-B
4.82/2.35	
4.82/2.35	     10: f2 -> f300 : A'=B, C'=0, [ A>=1+B ], cost: 1+A-B
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Computing asymptotic complexity for rule 8
4.82/2.35	
4.82/2.35	   Solved the limit problem by the following transformations:
4.82/2.35	
4.82/2.35	      Created initial limit problem:
4.82/2.35	
4.82/2.35	      free (+/+!), 1+A-B (+), A-B (+/+!) [not solved]
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	      removing all constraints (solved by SMT)
4.82/2.35	
4.82/2.35	      resulting limit problem: [solved]
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	      applying transformation rule (C) using substitution {free==1,A==0,B==-n}
4.82/2.35	
4.82/2.35	      resulting limit problem:
4.82/2.35	
4.82/2.35	      [solved]
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	   Solution:
4.82/2.35	
4.82/2.35	      free / 1
4.82/2.35	
4.82/2.35	      A / 0
4.82/2.35	
4.82/2.35	      B / -n
4.82/2.35	
4.82/2.35	   Resulting cost 1+n has complexity: Poly(n^1)
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Found new complexity Poly(n^1).
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	Obtained the following overall complexity (w.r.t. the length of the input n):
4.82/2.35	
4.82/2.35	   Complexity:  Poly(n^1)
4.82/2.35	
4.82/2.35	   Cpx degree:  1
4.82/2.35	
4.82/2.35	   Solved cost: 1+n
4.82/2.35	
4.82/2.35	   Rule cost:   1+A-B
4.82/2.35	
4.82/2.35	   Rule guard:  [ free>=1 && A>=1+B ]
4.82/2.35	
4.82/2.35	
4.82/2.35	
4.82/2.35	WORST_CASE(Omega(n^1),?)
4.82/2.35	
4.82/2.35	
4.82/2.35	----------------------------------------
4.82/2.35	
4.82/2.35	(4)
4.82/2.35	BOUNDS(n^1, INF)
4.82/2.38	EOF