5.53/2.93	WORST_CASE(Omega(n^1), ?)
5.59/2.93	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
5.59/2.93	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.59/2.93	
5.59/2.93	
5.59/2.93	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, INF).
5.59/2.93	
5.59/2.93	(0) CpxIntTrs
5.59/2.93	(1) Loat Proof [FINISHED, 1214 ms]
5.59/2.93	(2) BOUNDS(n^1, INF)
5.59/2.93	
5.59/2.93	
5.59/2.93	----------------------------------------
5.59/2.93	
5.59/2.93	(0)
5.59/2.93	Obligation:
5.59/2.93	Complexity Int TRS consisting of the following rules:
5.59/2.93	f2(A, B, C, D) -> Com_1(f300(A, B, C, D)) :|: TRUE
5.59/2.93	f300(A, B, C, D) -> Com_1(f300(-(1) + B, -(1) + B, E, D)) :|: A >= 1 && B >= 1 && E >= 1 && B + A >= 1
5.59/2.93	f300(A, B, C, D) -> Com_1(f300(-(1) + B, -(1) + B, E, D)) :|: A >= 1 && B >= 1 && 0 >= E + 1 && B + A >= 1
5.59/2.93	f300(A, B, C, D) -> Com_1(f300(-(1) + A, -(2) + A, 0, D)) :|: A >= 1 && B + A >= 1 && B >= 1
5.59/2.93	f300(A, B, C, D) -> Com_1(f1(A, B, C, E)) :|: A >= 1 && 0 >= B + A && B >= 1
5.59/2.93	f300(A, B, C, D) -> Com_1(f1(A, B, C, E)) :|: B >= 1 && 0 >= A
5.59/2.93	f300(A, B, C, D) -> Com_1(f1(A, B, C, E)) :|: 0 >= B
5.59/2.93	
5.59/2.93	The start-symbols are:[f2_4]
5.59/2.93	
5.59/2.93	
5.59/2.93	----------------------------------------
5.59/2.93	
5.59/2.93	(1) Loat Proof (FINISHED)
5.59/2.93	
5.59/2.93	
5.59/2.93	### Pre-processing the ITS problem ###
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	Initial linear ITS problem
5.59/2.93	
5.59/2.93	   Start location: f2
5.59/2.93	
5.59/2.93	      0: f2 -> f300 : [], cost: 1
5.59/2.93	
5.59/2.93	      1: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free, [ A>=1 && B>=1 && free>=1 && A+B>=1 ], cost: 1
5.59/2.93	
5.59/2.93	      2: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free_1, [ A>=1 && B>=1 && 0>=1+free_1 && A+B>=1 ], cost: 1
5.59/2.93	
5.59/2.93	      3: f300 -> f300 : A'=-1+A, B'=-2+A, C'=0, [ A>=1 && A+B>=1 && B>=1 ], cost: 1
5.59/2.93	
5.59/2.93	      4: f300 -> f1 : D'=free_2, [ A>=1 && 0>=A+B && B>=1 ], cost: 1
5.59/2.93	
5.59/2.93	      5: f300 -> f1 : D'=free_3, [ B>=1 && 0>=A ], cost: 1
5.59/2.93	
5.59/2.93	      6: f300 -> f1 : D'=free_4, [ 0>=B ], cost: 1
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	Removed unreachable and leaf rules:
5.59/2.93	
5.59/2.93	   Start location: f2
5.59/2.93	
5.59/2.93	      0: f2 -> f300 : [], cost: 1
5.59/2.93	
5.59/2.93	      1: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free, [ A>=1 && B>=1 && free>=1 && A+B>=1 ], cost: 1
5.59/2.93	
5.59/2.93	      2: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free_1, [ A>=1 && B>=1 && 0>=1+free_1 && A+B>=1 ], cost: 1
5.59/2.93	
5.59/2.93	      3: f300 -> f300 : A'=-1+A, B'=-2+A, C'=0, [ A>=1 && A+B>=1 && B>=1 ], cost: 1
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	### Simplification by acceleration and chaining ###
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	Accelerating simple loops of location 1.
5.59/2.93	
5.59/2.93	   Accelerating the following rules:
5.59/2.93	
5.59/2.93	      1: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free, [ A>=1 && B>=1 && free>=1 ], cost: 1
5.59/2.93	
5.59/2.93	      2: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free_1, [ A>=1 && B>=1 && 0>=1+free_1 ], cost: 1
5.59/2.93	
5.59/2.93	      3: f300 -> f300 : A'=-1+A, B'=-2+A, C'=0, [ A>=1 && A+B>=1 && B>=1 ], cost: 1
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	   Accelerated rule 1 with backward acceleration, yielding the new rule 7.
5.59/2.93	
5.59/2.93	   Accelerated rule 1 with backward acceleration, yielding the new rule 8.
5.59/2.93	
5.59/2.93	   Accelerated rule 2 with backward acceleration, yielding the new rule 9.
5.59/2.93	
5.59/2.93	   Accelerated rule 2 with backward acceleration, yielding the new rule 10.
5.59/2.93	
5.59/2.93	   Accelerated rule 3 with backward acceleration, yielding the new rule 11.
5.59/2.93	
5.59/2.93	   Accelerated rule 3 with backward acceleration, yielding the new rule 12.
5.59/2.93	
5.59/2.93	   Accelerated rule 3 with backward acceleration, yielding the new rule 13.
5.59/2.93	
5.59/2.93	   Removing the simple loops: 1 2 3.
5.59/2.93	
5.59/2.93	   Also removing duplicate rules:.
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	Accelerated all simple loops using metering functions (where possible):
5.59/2.93	
5.59/2.93	   Start location: f2
5.59/2.93	
5.59/2.93	      0: f2 -> f300 : [], cost: 1
5.59/2.93	
5.59/2.93	      8: f300 -> f300 : A'=0, B'=0, C'=free, [ A>=1 && B>=1 && free>=1 ], cost: B
5.59/2.93	
5.59/2.93	     10: f300 -> f300 : A'=0, B'=0, C'=free_1, [ A>=1 && B>=1 && 0>=1+free_1 ], cost: B
5.59/2.93	
5.59/2.93	     12: f300 -> f300 : A'=0, B'=-1, C'=0, [ 0>=1 ], cost: A
5.59/2.93	
5.59/2.93	     13: f300 -> f300 : A'=1, B'=0, C'=0, [ A+B>=1 && B>=1 && -1+A>0 ], cost: -1+A
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	Chained accelerated rules (with incoming rules):
5.59/2.93	
5.59/2.93	   Start location: f2
5.59/2.93	
5.59/2.93	      0: f2 -> f300 : [], cost: 1
5.59/2.93	
5.59/2.93	     14: f2 -> f300 : A'=0, B'=0, C'=free, [ A>=1 && B>=1 && free>=1 ], cost: 1+B
5.59/2.93	
5.59/2.93	     15: f2 -> f300 : A'=0, B'=0, C'=free_1, [ A>=1 && B>=1 && 0>=1+free_1 ], cost: 1+B
5.59/2.93	
5.59/2.93	     16: f2 -> f300 : A'=1, B'=0, C'=0, [ A+B>=1 && B>=1 && -1+A>0 ], cost: A
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	Removed unreachable locations (and leaf rules with constant cost):
5.59/2.93	
5.59/2.93	   Start location: f2
5.59/2.93	
5.59/2.93	     14: f2 -> f300 : A'=0, B'=0, C'=free, [ A>=1 && B>=1 && free>=1 ], cost: 1+B
5.59/2.93	
5.59/2.93	     15: f2 -> f300 : A'=0, B'=0, C'=free_1, [ A>=1 && B>=1 && 0>=1+free_1 ], cost: 1+B
5.59/2.93	
5.59/2.93	     16: f2 -> f300 : A'=1, B'=0, C'=0, [ A+B>=1 && B>=1 && -1+A>0 ], cost: A
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	### Computing asymptotic complexity ###
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	Fully simplified ITS problem
5.59/2.93	
5.59/2.93	   Start location: f2
5.59/2.93	
5.59/2.93	     14: f2 -> f300 : A'=0, B'=0, C'=free, [ A>=1 && B>=1 && free>=1 ], cost: 1+B
5.59/2.93	
5.59/2.93	     15: f2 -> f300 : A'=0, B'=0, C'=free_1, [ A>=1 && B>=1 && 0>=1+free_1 ], cost: 1+B
5.59/2.93	
5.59/2.93	     16: f2 -> f300 : A'=1, B'=0, C'=0, [ A+B>=1 && B>=1 && -1+A>0 ], cost: A
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	Computing asymptotic complexity for rule 14
5.59/2.93	
5.59/2.93	   Solved the limit problem by the following transformations:
5.59/2.93	
5.59/2.93	      Created initial limit problem:
5.59/2.93	
5.59/2.93	      free (+/+!), 1+B (+), A (+/+!), B (+/+!) [not solved]
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	      removing all constraints (solved by SMT)
5.59/2.93	
5.59/2.93	      resulting limit problem: [solved]
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	      applying transformation rule (C) using substitution {free==n,A==n,B==n}
5.59/2.93	
5.59/2.93	      resulting limit problem:
5.59/2.93	
5.59/2.93	      [solved]
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	   Solution:
5.59/2.93	
5.59/2.93	      free / n
5.59/2.93	
5.59/2.93	      A / n
5.59/2.93	
5.59/2.93	      B / n
5.59/2.93	
5.59/2.93	   Resulting cost 1+n has complexity: Poly(n^1)
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	Found new complexity Poly(n^1).
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	Obtained the following overall complexity (w.r.t. the length of the input n):
5.59/2.93	
5.59/2.93	   Complexity:  Poly(n^1)
5.59/2.93	
5.59/2.93	   Cpx degree:  1
5.59/2.93	
5.59/2.93	   Solved cost: 1+n
5.59/2.93	
5.59/2.93	   Rule cost:   1+B
5.59/2.93	
5.59/2.93	   Rule guard:  [ A>=1 && B>=1 && free>=1 ]
5.59/2.93	
5.59/2.93	
5.59/2.93	
5.59/2.93	WORST_CASE(Omega(n^1),?)
5.59/2.93	
5.59/2.93	
5.59/2.93	----------------------------------------
5.59/2.93	
5.59/2.93	(2)
5.59/2.93	BOUNDS(n^1, INF)
5.59/2.95	EOF