4.24/2.08	WORST_CASE(NON_POLY, ?)
4.24/2.08	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.24/2.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.24/2.08	
4.24/2.08	
4.24/2.08	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF).
4.24/2.08	
4.24/2.08	(0) CpxIntTrs
4.24/2.08	(1) Loat Proof [FINISHED, 421 ms]
4.24/2.08	(2) BOUNDS(INF, INF)
4.24/2.08	
4.24/2.08	
4.24/2.08	----------------------------------------
4.24/2.08	
4.24/2.08	(0)
4.24/2.08	Obligation:
4.24/2.08	Complexity Int TRS consisting of the following rules:
4.24/2.08	eval(A, B) -> Com_1(eval(A - 1, C)) :|: A >= 0
4.24/2.08	eval(A, B) -> Com_1(eval(A, B - 1)) :|: B >= 0
4.24/2.08	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
4.24/2.08	
4.24/2.08	The start-symbols are:[start_2]
4.24/2.08	
4.24/2.08	
4.24/2.08	----------------------------------------
4.24/2.08	
4.24/2.08	(1) Loat Proof (FINISHED)
4.24/2.08	
4.24/2.08	
4.24/2.08	### Pre-processing the ITS problem ###
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	Initial linear ITS problem
4.24/2.08	
4.24/2.08	   Start location: start
4.24/2.08	
4.24/2.08	      0: eval -> eval : A'=-1+A, B'=free, [ A>=0 ], cost: 1
4.24/2.08	
4.24/2.08	      1: eval -> eval : B'=-1+B, [ B>=0 ], cost: 1
4.24/2.08	
4.24/2.08	      2: start -> eval : [], cost: 1
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	### Simplification by acceleration and chaining ###
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	Accelerating simple loops of location 0.
4.24/2.08	
4.24/2.08	   Accelerating the following rules:
4.24/2.08	
4.24/2.08	      0: eval -> eval : A'=-1+A, B'=free, [ A>=0 ], cost: 1
4.24/2.08	
4.24/2.08	      1: eval -> eval : B'=-1+B, [ B>=0 ], cost: 1
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	   Accelerated rule 0 with metering function 1+A, yielding the new rule 3.
4.24/2.08	
4.24/2.08	   Accelerated rule 1 with metering function 1+B, yielding the new rule 4.
4.24/2.08	
4.24/2.08	   Nested simple loops 0 (outer loop) and 4 (inner loop) with metering function 1+A, resulting in the new rules: 5, 6.
4.24/2.08	
4.24/2.08	   Removing the simple loops: 0 1.
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	Accelerated all simple loops using metering functions (where possible):
4.24/2.08	
4.24/2.08	   Start location: start
4.24/2.08	
4.24/2.08	      3: eval -> eval : A'=-1, B'=free, [ A>=0 ], cost: 1+A
4.24/2.08	
4.24/2.08	      4: eval -> eval : B'=-1, [ B>=0 ], cost: 1+B
4.24/2.08	
4.24/2.08	      5: eval -> eval : A'=-1, B'=-1, [ A>=0 && free>=0 ], cost: 2+free*(1+A)+2*A
4.24/2.08	
4.24/2.08	      6: eval -> eval : A'=-1, B'=-1, [ B>=0 && A>=0 && free>=0 ], cost: 3+free*(1+A)+2*A+B
4.24/2.08	
4.24/2.08	      2: start -> eval : [], cost: 1
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	Chained accelerated rules (with incoming rules):
4.24/2.08	
4.24/2.08	   Start location: start
4.24/2.08	
4.24/2.08	      2: start -> eval : [], cost: 1
4.24/2.08	
4.24/2.08	      7: start -> eval : A'=-1, B'=free, [ A>=0 ], cost: 2+A
4.24/2.08	
4.24/2.08	      8: start -> eval : B'=-1, [ B>=0 ], cost: 2+B
4.24/2.08	
4.24/2.08	      9: start -> eval : A'=-1, B'=-1, [ A>=0 && free>=0 ], cost: 3+free*(1+A)+2*A
4.24/2.08	
4.24/2.08	     10: start -> eval : A'=-1, B'=-1, [ B>=0 && A>=0 && free>=0 ], cost: 4+free*(1+A)+2*A+B
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	Removed unreachable locations (and leaf rules with constant cost):
4.24/2.08	
4.24/2.08	   Start location: start
4.24/2.08	
4.24/2.08	      7: start -> eval : A'=-1, B'=free, [ A>=0 ], cost: 2+A
4.24/2.08	
4.24/2.08	      8: start -> eval : B'=-1, [ B>=0 ], cost: 2+B
4.24/2.08	
4.24/2.08	      9: start -> eval : A'=-1, B'=-1, [ A>=0 && free>=0 ], cost: 3+free*(1+A)+2*A
4.24/2.08	
4.24/2.08	     10: start -> eval : A'=-1, B'=-1, [ B>=0 && A>=0 && free>=0 ], cost: 4+free*(1+A)+2*A+B
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	### Computing asymptotic complexity ###
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	Fully simplified ITS problem
4.24/2.08	
4.24/2.08	   Start location: start
4.24/2.08	
4.24/2.08	      7: start -> eval : A'=-1, B'=free, [ A>=0 ], cost: 2+A
4.24/2.08	
4.24/2.08	      8: start -> eval : B'=-1, [ B>=0 ], cost: 2+B
4.24/2.08	
4.24/2.08	      9: start -> eval : A'=-1, B'=-1, [ A>=0 && free>=0 ], cost: 3+free*(1+A)+2*A
4.24/2.08	
4.24/2.08	     10: start -> eval : A'=-1, B'=-1, [ B>=0 && A>=0 && free>=0 ], cost: 4+free*(1+A)+2*A+B
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	Computing asymptotic complexity for rule 7
4.24/2.08	
4.24/2.08	   Solved the limit problem by the following transformations:
4.24/2.08	
4.24/2.08	      Created initial limit problem:
4.24/2.08	
4.24/2.08	      2+A (+), 1+A (+/+!) [not solved]
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	      removing all constraints (solved by SMT)
4.24/2.08	
4.24/2.08	      resulting limit problem: [solved]
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	      applying transformation rule (C) using substitution {A==n}
4.24/2.08	
4.24/2.08	      resulting limit problem:
4.24/2.08	
4.24/2.08	      [solved]
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	   Solution:
4.24/2.08	
4.24/2.08	      A / n
4.24/2.08	
4.24/2.08	   Resulting cost 2+n has complexity: Poly(n^1)
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	Found new complexity Poly(n^1).
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	Computing asymptotic complexity for rule 9
4.24/2.08	
4.24/2.08	   Solved the limit problem by the following transformations:
4.24/2.08	
4.24/2.08	      Created initial limit problem:
4.24/2.08	
4.24/2.08	      3+free+2*A+free*A (+), 1+free (+/+!), 1+A (+/+!) [not solved]
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	      removing all constraints (solved by SMT)
4.24/2.08	
4.24/2.08	      resulting limit problem: [solved]
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	      applying transformation rule (C) using substitution {free==n,A==0}
4.24/2.08	
4.24/2.08	      resulting limit problem:
4.24/2.08	
4.24/2.08	      [solved]
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	   Solution:
4.24/2.08	
4.24/2.08	      free / n
4.24/2.08	
4.24/2.08	      A / 0
4.24/2.08	
4.24/2.08	   Resulting cost 3+n has complexity: Unbounded
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	Found new complexity Unbounded.
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	Obtained the following overall complexity (w.r.t. the length of the input n):
4.24/2.08	
4.24/2.08	   Complexity:  Unbounded
4.24/2.08	
4.24/2.08	   Cpx degree:  Unbounded
4.24/2.08	
4.24/2.08	   Solved cost: 3+n
4.24/2.08	
4.24/2.08	   Rule cost:   3+free*(1+A)+2*A
4.24/2.08	
4.24/2.08	   Rule guard:  [ A>=0 && free>=0 ]
4.24/2.08	
4.24/2.08	
4.24/2.08	
4.24/2.08	WORST_CASE(INF,?)
4.24/2.08	
4.24/2.08	
4.24/2.08	----------------------------------------
4.24/2.08	
4.24/2.08	(2)
4.24/2.08	BOUNDS(INF, INF)
4.41/2.12	EOF