5.22/2.55	WORST_CASE(NON_POLY, ?)
5.22/2.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
5.22/2.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.22/2.55	
5.22/2.55	
5.22/2.55	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF).
5.22/2.55	
5.22/2.55	(0) CpxIntTrs
5.22/2.55	(1) Loat Proof [FINISHED, 836 ms]
5.22/2.55	(2) BOUNDS(INF, INF)
5.22/2.55	
5.22/2.55	
5.22/2.55	----------------------------------------
5.22/2.55	
5.22/2.55	(0)
5.22/2.55	Obligation:
5.22/2.55	Complexity Int TRS consisting of the following rules:
5.22/2.55	f0(A, B) -> Com_1(f1(C, 1)) :|: TRUE
5.22/2.55	f1(A, B) -> Com_1(f1(A - 10, B - 1)) :|: B >= 1 && A >= 101
5.22/2.55	f1(A, B) -> Com_1(f1(A + 11, B + 1)) :|: B >= 1 && 100 >= A
5.22/2.55	
5.22/2.55	The start-symbols are:[f0_2]
5.22/2.55	
5.22/2.55	
5.22/2.55	----------------------------------------
5.22/2.55	
5.22/2.55	(1) Loat Proof (FINISHED)
5.22/2.55	
5.22/2.55	
5.22/2.55	### Pre-processing the ITS problem ###
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	Initial linear ITS problem
5.22/2.55	
5.22/2.55	   Start location: f0
5.22/2.55	
5.22/2.55	      0: f0 -> f1 : A'=free, B'=1, [], cost: 1
5.22/2.55	
5.22/2.55	      1: f1 -> f1 : A'=-10+A, B'=-1+B, [ B>=1 && A>=101 ], cost: 1
5.22/2.55	
5.22/2.55	      2: f1 -> f1 : A'=11+A, B'=1+B, [ B>=1 && 100>=A ], cost: 1
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	### Simplification by acceleration and chaining ###
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	Accelerating simple loops of location 1.
5.22/2.55	
5.22/2.55	   Accelerating the following rules:
5.22/2.55	
5.22/2.55	      1: f1 -> f1 : A'=-10+A, B'=-1+B, [ B>=1 && A>=101 ], cost: 1
5.22/2.55	
5.22/2.55	      2: f1 -> f1 : A'=11+A, B'=1+B, [ B>=1 && 100>=A ], cost: 1
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	   Accelerated rule 1 with metering function meter (where 10*meter==-100+A) (after adding A<=B), yielding the new rule 3.
5.22/2.55	
5.22/2.55	   Accelerated rule 1 with backward acceleration, yielding the new rule 4.
5.22/2.55	
5.22/2.55	   Accelerated rule 2 with metering function meter_1 (where 11*meter_1==100-A), yielding the new rule 5.
5.22/2.55	
5.22/2.55	      During metering: Instantiating temporary variables by {meter==1}
5.22/2.55	
5.22/2.55	   Nested simple loops 2 (outer loop) and 3 (inner loop) with metering function 89+10*meter-A, resulting in the new rules: 6.
5.22/2.55	
5.22/2.55	      During metering: Instantiating temporary variables by {k==-1+B}
5.22/2.55	
5.22/2.55	      During metering: Instantiating temporary variables by {k==1}
5.22/2.55	
5.22/2.55	   Removing the simple loops: 1 2.
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	Accelerated all simple loops using metering functions (where possible):
5.22/2.55	
5.22/2.55	   Start location: f0
5.22/2.55	
5.22/2.55	      0: f0 -> f1 : A'=free, B'=1, [], cost: 1
5.22/2.55	
5.22/2.55	      3: f1 -> f1 : A'=-10*meter+A, B'=-meter+B, [ B>=1 && A>=101 && A<=B && 10*meter==-100+A && meter>=1 ], cost: meter
5.22/2.55	
5.22/2.55	      4: f1 -> f1 : A'=A-10*k, B'=-k+B, [ B>=1 && A>=101 && k>0 && 1-k+B>=1 && 10+A-10*k>=101 ], cost: k
5.22/2.55	
5.22/2.55	      5: f1 -> f1 : A'=11*meter_1+A, B'=meter_1+B, [ B>=1 && 100>=A && 11*meter_1==100-A && meter_1>=1 ], cost: meter_1
5.22/2.55	
5.22/2.55	      6: f1 -> f1 : A'=979-10*(89+10*meter-A)*meter+110*meter-10*A, B'=89-(89+10*meter-A)*meter+10*meter-A+B, [ B>=1 && 100>=A && 11+A>=101 && 11+A<=1+B && 10*meter==-89+A && meter>=1 && 89+10*meter-A>=1 ], cost: 89+(89+10*meter-A)*meter+10*meter-A
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	Chained accelerated rules (with incoming rules):
5.22/2.55	
5.22/2.55	   Start location: f0
5.22/2.55	
5.22/2.55	      0: f0 -> f1 : A'=free, B'=1, [], cost: 1
5.22/2.55	
5.22/2.55	      7: f0 -> f1 : A'=-10+free, B'=0, [ free>=101 ], cost: 2
5.22/2.55	
5.22/2.55	      8: f0 -> f1 : A'=100, B'=1+meter_1, [ 100>=100-11*meter_1 && meter_1>=1 ], cost: 1+meter_1
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	Removed unreachable locations (and leaf rules with constant cost):
5.22/2.55	
5.22/2.55	   Start location: f0
5.22/2.55	
5.22/2.55	      8: f0 -> f1 : A'=100, B'=1+meter_1, [ 100>=100-11*meter_1 && meter_1>=1 ], cost: 1+meter_1
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	### Computing asymptotic complexity ###
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	Fully simplified ITS problem
5.22/2.55	
5.22/2.55	   Start location: f0
5.22/2.55	
5.22/2.55	      8: f0 -> f1 : A'=100, B'=1+meter_1, [ 100>=100-11*meter_1 && meter_1>=1 ], cost: 1+meter_1
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	Computing asymptotic complexity for rule 8
5.22/2.55	
5.22/2.55	   Simplified the guard:
5.22/2.55	
5.22/2.55	      8: f0 -> f1 : A'=100, B'=1+meter_1, [ meter_1>=1 ], cost: 1+meter_1
5.22/2.55	
5.22/2.55	   Solved the limit problem by the following transformations:
5.22/2.55	
5.22/2.55	      Created initial limit problem:
5.22/2.55	
5.22/2.55	      1+meter_1 (+), meter_1 (+/+!) [not solved]
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	      removing all constraints (solved by SMT)
5.22/2.55	
5.22/2.55	      resulting limit problem: [solved]
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	      applying transformation rule (C) using substitution {meter_1==n}
5.22/2.55	
5.22/2.55	      resulting limit problem:
5.22/2.55	
5.22/2.55	      [solved]
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	   Solution:
5.22/2.55	
5.22/2.55	      meter_1 / n
5.22/2.55	
5.22/2.55	   Resulting cost 1+n has complexity: Unbounded
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	Found new complexity Unbounded.
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	Obtained the following overall complexity (w.r.t. the length of the input n):
5.22/2.55	
5.22/2.55	   Complexity:  Unbounded
5.22/2.55	
5.22/2.55	   Cpx degree:  Unbounded
5.22/2.55	
5.22/2.55	   Solved cost: 1+n
5.22/2.55	
5.22/2.55	   Rule cost:   1+meter_1
5.22/2.55	
5.22/2.55	   Rule guard:  [ meter_1>=1 ]
5.22/2.55	
5.22/2.55	
5.22/2.55	
5.22/2.55	WORST_CASE(INF,?)
5.22/2.55	
5.22/2.55	
5.22/2.55	----------------------------------------
5.22/2.55	
5.22/2.55	(2)
5.22/2.55	BOUNDS(INF, INF)
5.33/2.58	EOF