3.77/1.80	WORST_CASE(Omega(n^1), O(n^1))
3.81/2.35	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.81/2.35	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.81/2.35	
3.81/2.35	
3.81/2.35	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
3.81/2.35	
3.81/2.35	(0) CpxIntTrs
3.81/2.35	(1) Koat Proof [FINISHED, 5 ms]
3.81/2.35	(2) BOUNDS(1, n^1)
3.81/2.35	(3) Loat Proof [FINISHED, 134 ms]
3.81/2.35	(4) BOUNDS(n^1, INF)
3.81/2.35	
3.81/2.35	
3.81/2.35	----------------------------------------
3.81/2.35	
3.81/2.35	(0)
3.81/2.35	Obligation:
3.81/2.35	Complexity Int TRS consisting of the following rules:
3.81/2.35	sumto(A, B) -> Com_1(end(A, B)) :|: A >= B + 1
3.81/2.35	sumto(A, B) -> Com_1(sumto(A + 1, B)) :|: B >= A
3.81/2.35	start(A, B) -> Com_1(sumto(A, B)) :|: TRUE
3.81/2.35	
3.81/2.35	The start-symbols are:[start_2]
3.81/2.35	
3.81/2.35	
3.81/2.35	----------------------------------------
3.81/2.35	
3.81/2.35	(1) Koat Proof (FINISHED)
3.81/2.35	YES(?, ar_0 + ar_1 + 3)
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Initial complexity problem:
3.81/2.35	
3.81/2.35	1:	T:
3.81/2.35	
3.81/2.35			(Comp: ?, Cost: 1)    sumto(ar_0, ar_1) -> Com_1(end(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
3.81/2.35	
3.81/2.35			(Comp: ?, Cost: 1)    sumto(ar_0, ar_1) -> Com_1(sumto(ar_0 + 1, ar_1)) [ ar_1 >= ar_0 ]
3.81/2.35	
3.81/2.35			(Comp: ?, Cost: 1)    start(ar_0, ar_1) -> Com_1(sumto(ar_0, ar_1))
3.81/2.35	
3.81/2.35			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(start(ar_0, ar_1)) [ 0 <= 0 ]
3.81/2.35	
3.81/2.35		start location:	koat_start
3.81/2.35	
3.81/2.35		leaf cost:	0
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.81/2.35	
3.81/2.35	2:	T:
3.81/2.35	
3.81/2.35			(Comp: ?, Cost: 1)    sumto(ar_0, ar_1) -> Com_1(end(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
3.81/2.35	
3.81/2.35			(Comp: ?, Cost: 1)    sumto(ar_0, ar_1) -> Com_1(sumto(ar_0 + 1, ar_1)) [ ar_1 >= ar_0 ]
3.81/2.35	
3.81/2.35			(Comp: 1, Cost: 1)    start(ar_0, ar_1) -> Com_1(sumto(ar_0, ar_1))
3.81/2.35	
3.81/2.35			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(start(ar_0, ar_1)) [ 0 <= 0 ]
3.81/2.35	
3.81/2.35		start location:	koat_start
3.81/2.35	
3.81/2.35		leaf cost:	0
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	A polynomial rank function with
3.81/2.35	
3.81/2.35		Pol(sumto) = 1
3.81/2.35	
3.81/2.35		Pol(end) = 0
3.81/2.35	
3.81/2.35		Pol(start) = 1
3.81/2.35	
3.81/2.35		Pol(koat_start) = 1
3.81/2.35	
3.81/2.35	orients all transitions weakly and the transition
3.81/2.35	
3.81/2.35		sumto(ar_0, ar_1) -> Com_1(end(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
3.81/2.35	
3.81/2.35	strictly and produces the following problem:
3.81/2.35	
3.81/2.35	3:	T:
3.81/2.35	
3.81/2.35			(Comp: 1, Cost: 1)    sumto(ar_0, ar_1) -> Com_1(end(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
3.81/2.35	
3.81/2.35			(Comp: ?, Cost: 1)    sumto(ar_0, ar_1) -> Com_1(sumto(ar_0 + 1, ar_1)) [ ar_1 >= ar_0 ]
3.81/2.35	
3.81/2.35			(Comp: 1, Cost: 1)    start(ar_0, ar_1) -> Com_1(sumto(ar_0, ar_1))
3.81/2.35	
3.81/2.35			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(start(ar_0, ar_1)) [ 0 <= 0 ]
3.81/2.35	
3.81/2.35		start location:	koat_start
3.81/2.35	
3.81/2.35		leaf cost:	0
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	A polynomial rank function with
3.81/2.35	
3.81/2.35		Pol(sumto) = -V_1 + V_2 + 1
3.81/2.35	
3.81/2.35		Pol(end) = -V_1 + V_2
3.81/2.35	
3.81/2.35		Pol(start) = -V_1 + V_2 + 1
3.81/2.35	
3.81/2.35		Pol(koat_start) = -V_1 + V_2 + 1
3.81/2.35	
3.81/2.35	orients all transitions weakly and the transition
3.81/2.35	
3.81/2.35		sumto(ar_0, ar_1) -> Com_1(sumto(ar_0 + 1, ar_1)) [ ar_1 >= ar_0 ]
3.81/2.35	
3.81/2.35	strictly and produces the following problem:
3.81/2.35	
3.81/2.35	4:	T:
3.81/2.35	
3.81/2.35			(Comp: 1, Cost: 1)                  sumto(ar_0, ar_1) -> Com_1(end(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
3.81/2.35	
3.81/2.35			(Comp: ar_0 + ar_1 + 1, Cost: 1)    sumto(ar_0, ar_1) -> Com_1(sumto(ar_0 + 1, ar_1)) [ ar_1 >= ar_0 ]
3.81/2.35	
3.81/2.35			(Comp: 1, Cost: 1)                  start(ar_0, ar_1) -> Com_1(sumto(ar_0, ar_1))
3.81/2.35	
3.81/2.35			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1) -> Com_1(start(ar_0, ar_1)) [ 0 <= 0 ]
3.81/2.35	
3.81/2.35		start location:	koat_start
3.81/2.35	
3.81/2.35		leaf cost:	0
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Complexity upper bound ar_0 + ar_1 + 3
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Time: 0.030 sec (SMT: 0.027 sec)
3.81/2.35	
3.81/2.35	
3.81/2.35	----------------------------------------
3.81/2.35	
3.81/2.35	(2)
3.81/2.35	BOUNDS(1, n^1)
3.81/2.35	
3.81/2.35	----------------------------------------
3.81/2.35	
3.81/2.35	(3) Loat Proof (FINISHED)
3.81/2.35	
3.81/2.35	
3.81/2.35	### Pre-processing the ITS problem ###
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Initial linear ITS problem
3.81/2.35	
3.81/2.35	   Start location: start
3.81/2.35	
3.81/2.35	      0: sumto -> end : [ A>=1+B ], cost: 1
3.81/2.35	
3.81/2.35	      1: sumto -> sumto : A'=1+A, [ B>=A ], cost: 1
3.81/2.35	
3.81/2.35	      2: start -> sumto : [], cost: 1
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Removed unreachable and leaf rules:
3.81/2.35	
3.81/2.35	   Start location: start
3.81/2.35	
3.81/2.35	      1: sumto -> sumto : A'=1+A, [ B>=A ], cost: 1
3.81/2.35	
3.81/2.35	      2: start -> sumto : [], cost: 1
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	### Simplification by acceleration and chaining ###
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Accelerating simple loops of location 0.
3.81/2.35	
3.81/2.35	   Accelerating the following rules:
3.81/2.35	
3.81/2.35	      1: sumto -> sumto : A'=1+A, [ B>=A ], cost: 1
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	   Accelerated rule 1 with metering function 1-A+B, yielding the new rule 3.
3.81/2.35	
3.81/2.35	   Removing the simple loops: 1.
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Accelerated all simple loops using metering functions (where possible):
3.81/2.35	
3.81/2.35	   Start location: start
3.81/2.35	
3.81/2.35	      3: sumto -> sumto : A'=1+B, [ B>=A ], cost: 1-A+B
3.81/2.35	
3.81/2.35	      2: start -> sumto : [], cost: 1
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Chained accelerated rules (with incoming rules):
3.81/2.35	
3.81/2.35	   Start location: start
3.81/2.35	
3.81/2.35	      2: start -> sumto : [], cost: 1
3.81/2.35	
3.81/2.35	      4: start -> sumto : A'=1+B, [ B>=A ], cost: 2-A+B
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Removed unreachable locations (and leaf rules with constant cost):
3.81/2.35	
3.81/2.35	   Start location: start
3.81/2.35	
3.81/2.35	      4: start -> sumto : A'=1+B, [ B>=A ], cost: 2-A+B
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	### Computing asymptotic complexity ###
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Fully simplified ITS problem
3.81/2.35	
3.81/2.35	   Start location: start
3.81/2.35	
3.81/2.35	      4: start -> sumto : A'=1+B, [ B>=A ], cost: 2-A+B
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Computing asymptotic complexity for rule 4
3.81/2.35	
3.81/2.35	   Solved the limit problem by the following transformations:
3.81/2.35	
3.81/2.35	      Created initial limit problem:
3.81/2.35	
3.81/2.35	      1-A+B (+/+!), 2-A+B (+) [not solved]
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	      removing all constraints (solved by SMT)
3.81/2.35	
3.81/2.35	      resulting limit problem: [solved]
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	      applying transformation rule (C) using substitution {A==0,B==n}
3.81/2.35	
3.81/2.35	      resulting limit problem:
3.81/2.35	
3.81/2.35	      [solved]
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	   Solution:
3.81/2.35	
3.81/2.35	      A / 0
3.81/2.35	
3.81/2.35	      B / n
3.81/2.35	
3.81/2.35	   Resulting cost 2+n has complexity: Poly(n^1)
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Found new complexity Poly(n^1).
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	Obtained the following overall complexity (w.r.t. the length of the input n):
3.81/2.35	
3.81/2.35	   Complexity:  Poly(n^1)
3.81/2.35	
3.81/2.35	   Cpx degree:  1
3.81/2.35	
3.81/2.35	   Solved cost: 2+n
3.81/2.35	
3.81/2.35	   Rule cost:   2-A+B
3.81/2.35	
3.81/2.35	   Rule guard:  [ B>=A ]
3.81/2.35	
3.81/2.35	
3.81/2.35	
3.81/2.35	WORST_CASE(Omega(n^1),?)
3.81/2.35	
3.81/2.35	
3.81/2.35	----------------------------------------
3.81/2.35	
3.81/2.35	(4)
3.81/2.35	BOUNDS(n^1, INF)
3.83/2.41	EOF