4.95/2.39	WORST_CASE(Omega(n^1), O(n^1))
4.95/2.40	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.95/2.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.95/2.40	
4.95/2.40	
4.95/2.40	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + -1 * Arg_1) + nat(Arg_0 + -1 * Arg_0 * Arg_1 + -1 * Arg_2 * Arg_0 + Arg_0^2 + Arg_2 * Arg_1 + -1 * Arg_2) + nat(Arg_0 + -1 * Arg_2) + nat(Arg_0 + -1 * Arg_1) + nat(1 + Arg_0 + -1 * Arg_1)).
4.95/2.40	
4.95/2.40	(0) CpxIntTrs
4.95/2.40	(1) Loat Proof [FINISHED, 654 ms]
4.95/2.40	(2) BOUNDS(n^1, INF)
4.95/2.40	
4.95/2.40	
4.95/2.40	----------------------------------------
4.95/2.40	
4.95/2.40	(0)
4.95/2.40	Obligation:
4.95/2.40	Complexity Int TRS consisting of the following rules:
4.95/2.40	eval1(A, B, C) -> Com_1(eval2(A, B, C)) :|: A >= B + 1
4.95/2.40	eval2(A, B, C) -> Com_1(eval1(A, B + 1, C)) :|: A >= C + 1
4.95/2.40	eval2(A, B, C) -> Com_1(eval1(A, B, C + 1)) :|: A >= C + 1
4.95/2.40	eval2(A, B, C) -> Com_1(eval1(A - 1, B, C)) :|: C >= A
4.95/2.40	start(A, B, C) -> Com_1(eval1(A, B, C)) :|: TRUE
4.95/2.40	
4.95/2.40	The start-symbols are:[start_3]
4.95/2.40	
4.95/2.40	
4.95/2.40	----------------------------------------
4.95/2.40	
4.95/2.40	(1) Loat Proof (FINISHED)
4.95/2.40	
4.95/2.40	
4.95/2.40	### Pre-processing the ITS problem ###
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	Initial linear ITS problem
4.95/2.40	
4.95/2.40	   Start location: start
4.95/2.40	
4.95/2.40	      0: eval1 -> eval2 : [ A>=1+B ], cost: 1
4.95/2.40	
4.95/2.40	      1: eval2 -> eval1 : B'=1+B, [ A>=1+C ], cost: 1
4.95/2.40	
4.95/2.40	      2: eval2 -> eval1 : C'=1+C, [ A>=1+C ], cost: 1
4.95/2.40	
4.95/2.40	      3: eval2 -> eval1 : A'=-1+A, [ C>=A ], cost: 1
4.95/2.40	
4.95/2.40	      4: start -> eval1 : [], cost: 1
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	### Simplification by acceleration and chaining ###
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	Eliminated locations (on tree-shaped paths):
4.95/2.40	
4.95/2.40	   Start location: start
4.95/2.40	
4.95/2.40	      5: eval1 -> eval1 : B'=1+B, [ A>=1+B && A>=1+C ], cost: 2
4.95/2.40	
4.95/2.40	      6: eval1 -> eval1 : C'=1+C, [ A>=1+B && A>=1+C ], cost: 2
4.95/2.40	
4.95/2.40	      7: eval1 -> eval1 : A'=-1+A, [ A>=1+B && C>=A ], cost: 2
4.95/2.40	
4.95/2.40	      4: start -> eval1 : [], cost: 1
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	Accelerating simple loops of location 0.
4.95/2.40	
4.95/2.40	   Accelerating the following rules:
4.95/2.40	
4.95/2.40	      5: eval1 -> eval1 : B'=1+B, [ A>=1+B && A>=1+C ], cost: 2
4.95/2.40	
4.95/2.40	      6: eval1 -> eval1 : C'=1+C, [ A>=1+B && A>=1+C ], cost: 2
4.95/2.40	
4.95/2.40	      7: eval1 -> eval1 : A'=-1+A, [ A>=1+B && C>=A ], cost: 2
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	   Accelerated rule 5 with metering function A-B, yielding the new rule 8.
4.95/2.40	
4.95/2.40	   Accelerated rule 6 with metering function -C+A, yielding the new rule 9.
4.95/2.40	
4.95/2.40	   Accelerated rule 7 with metering function A-B, yielding the new rule 10.
4.95/2.40	
4.95/2.40	   Removing the simple loops: 5 6 7.
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	Accelerated all simple loops using metering functions (where possible):
4.95/2.40	
4.95/2.40	   Start location: start
4.95/2.40	
4.95/2.40	      8: eval1 -> eval1 : B'=A, [ A>=1+B && A>=1+C ], cost: 2*A-2*B
4.95/2.40	
4.95/2.40	      9: eval1 -> eval1 : C'=A, [ A>=1+B && A>=1+C ], cost: -2*C+2*A
4.95/2.40	
4.95/2.40	     10: eval1 -> eval1 : A'=B, [ A>=1+B && C>=A ], cost: 2*A-2*B
4.95/2.40	
4.95/2.40	      4: start -> eval1 : [], cost: 1
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	Chained accelerated rules (with incoming rules):
4.95/2.40	
4.95/2.40	   Start location: start
4.95/2.40	
4.95/2.40	      4: start -> eval1 : [], cost: 1
4.95/2.40	
4.95/2.40	     11: start -> eval1 : B'=A, [ A>=1+B && A>=1+C ], cost: 1+2*A-2*B
4.95/2.40	
4.95/2.40	     12: start -> eval1 : C'=A, [ A>=1+B && A>=1+C ], cost: 1-2*C+2*A
4.95/2.40	
4.95/2.40	     13: start -> eval1 : A'=B, [ A>=1+B && C>=A ], cost: 1+2*A-2*B
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	Removed unreachable locations (and leaf rules with constant cost):
4.95/2.40	
4.95/2.40	   Start location: start
4.95/2.40	
4.95/2.40	     11: start -> eval1 : B'=A, [ A>=1+B && A>=1+C ], cost: 1+2*A-2*B
4.95/2.40	
4.95/2.40	     12: start -> eval1 : C'=A, [ A>=1+B && A>=1+C ], cost: 1-2*C+2*A
4.95/2.40	
4.95/2.40	     13: start -> eval1 : A'=B, [ A>=1+B && C>=A ], cost: 1+2*A-2*B
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	### Computing asymptotic complexity ###
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	Fully simplified ITS problem
4.95/2.40	
4.95/2.40	   Start location: start
4.95/2.40	
4.95/2.40	     11: start -> eval1 : B'=A, [ A>=1+B && A>=1+C ], cost: 1+2*A-2*B
4.95/2.40	
4.95/2.40	     12: start -> eval1 : C'=A, [ A>=1+B && A>=1+C ], cost: 1-2*C+2*A
4.95/2.40	
4.95/2.40	     13: start -> eval1 : A'=B, [ A>=1+B && C>=A ], cost: 1+2*A-2*B
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	Computing asymptotic complexity for rule 11
4.95/2.40	
4.95/2.40	   Solved the limit problem by the following transformations:
4.95/2.40	
4.95/2.40	      Created initial limit problem:
4.95/2.40	
4.95/2.40	      -C+A (+/+!), A-B (+/+!), 1+2*A-2*B (+) [not solved]
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	      removing all constraints (solved by SMT)
4.95/2.40	
4.95/2.40	      resulting limit problem: [solved]
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	      applying transformation rule (C) using substitution {C==0,A==1,B==-n}
4.95/2.40	
4.95/2.40	      resulting limit problem:
4.95/2.40	
4.95/2.40	      [solved]
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	   Solution:
4.95/2.40	
4.95/2.40	      C / 0
4.95/2.40	
4.95/2.40	      A / 1
4.95/2.40	
4.95/2.40	      B / -n
4.95/2.40	
4.95/2.40	   Resulting cost 3+2*n has complexity: Poly(n^1)
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	Found new complexity Poly(n^1).
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	Obtained the following overall complexity (w.r.t. the length of the input n):
4.95/2.40	
4.95/2.40	   Complexity:  Poly(n^1)
4.95/2.40	
4.95/2.40	   Cpx degree:  1
4.95/2.40	
4.95/2.40	   Solved cost: 3+2*n
4.95/2.40	
4.95/2.40	   Rule cost:   1+2*A-2*B
4.95/2.40	
4.95/2.40	   Rule guard:  [ A>=1+B && A>=1+C ]
4.95/2.40	
4.95/2.40	
4.95/2.40	
4.95/2.40	WORST_CASE(Omega(n^1),?)
4.95/2.40	
4.95/2.40	
4.95/2.40	----------------------------------------
4.95/2.40	
4.95/2.40	(2)
4.95/2.40	BOUNDS(n^1, INF)
5.04/2.42	EOF