20.75/16.42	MAYBE
20.75/16.43	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
20.75/16.43	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.75/16.43	
20.75/16.43	
20.75/16.43	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, INF).
20.75/16.43	
20.75/16.43	(0) CpxIntTrs
20.75/16.43	(1) Loat Proof [FINISHED, 14.8 s]
20.75/16.43	(2) BOUNDS(1, INF)
20.75/16.43	
20.75/16.43	
20.75/16.43	----------------------------------------
20.75/16.43	
20.75/16.43	(0)
20.75/16.43	Obligation:
20.75/16.43	Complexity Int TRS consisting of the following rules:
20.75/16.43	eval(A, B, C) -> Com_1(eval(A + 1, B + A, C)) :|: A >= B
20.75/16.43	eval(A, B, C) -> Com_1(eval(A - C, B + C * C, C - 1)) :|: A >= B
20.75/16.43	start(A, B, C) -> Com_1(eval(A, B, C)) :|: TRUE
20.75/16.43	
20.75/16.43	The start-symbols are:[start_3]
20.75/16.43	
20.75/16.43	
20.75/16.43	----------------------------------------
20.75/16.43	
20.75/16.43	(1) Loat Proof (FINISHED)
20.75/16.43	
20.75/16.43	
20.75/16.43	### Pre-processing the ITS problem ###
20.75/16.43	
20.75/16.43	
20.75/16.43	
20.75/16.43	Initial linear ITS problem
20.75/16.43	
20.75/16.43	   Start location: start
20.75/16.43	
20.75/16.43	      0: eval -> eval : A'=1+A, B'=A+B, [ A>=B ], cost: 1
20.75/16.43	
20.75/16.43	      1: eval -> eval : A'=-C+A, B'=C^2+B, C'=-1+C, [ A>=B ], cost: 1
20.75/16.43	
20.75/16.43	      2: start -> eval : [], cost: 1
20.75/16.43	
20.75/16.43	
20.75/16.43	
20.75/16.43	### Simplification by acceleration and chaining ###
20.75/16.43	
20.75/16.43	
20.75/16.43	
20.75/16.43	Accelerating simple loops of location 0.
20.75/16.43	
20.75/16.43	   Accelerating the following rules:
20.75/16.43	
20.75/16.43	      0: eval -> eval : A'=1+A, B'=A+B, [ A>=B ], cost: 1
20.75/16.43	
20.75/16.43	      1: eval -> eval : A'=-C+A, B'=C^2+B, C'=-1+C, [ A>=B ], cost: 1
20.75/16.43	
20.75/16.43	
20.75/16.43	
20.75/16.43	   Found no metering function for rule 0.
20.75/16.43	
20.75/16.43	   Accelerated rule 1 with backward acceleration, yielding the new rule 3.
20.75/16.43	
20.75/16.43	   Removing the simple loops: 1.
20.75/16.43	
20.75/16.43	
20.75/16.43	
20.75/16.43	Accelerated all simple loops using metering functions (where possible):
20.75/16.43	
20.75/16.43	   Start location: start
20.75/16.43	
20.75/16.43	      0: eval -> eval : A'=1+A, B'=A+B, [ A>=B ], cost: 1
20.75/16.43	
20.75/16.43	      3: eval -> eval : A'=-1+1/2*k+A-k*C+1/2*k^2, B'=1-k^2*C-11/6*k+2*C+k*C^2+1/3*k^3-k*C+1/2*k^2+B, C'=-k+C, [ A>=B && k>0 && -3/2-(-1+k)*C+1/2*k+1/2*(-1+k)^2+A>=17/6-(-1+k)*C-11/6*k+2*C+(-1+k)*C^2+1/2*(-1+k)^2+1/3*(-1+k)^3-(-1+k)^2*C+B ], cost: k
20.75/16.43	
20.75/16.43	      2: start -> eval : [], cost: 1
20.75/16.43	
20.75/16.43	
20.75/16.43	
20.75/16.43	Chained accelerated rules (with incoming rules):
20.75/16.43	
20.75/16.43	   Start location: start
20.75/16.43	
20.75/16.43	      2: start -> eval : [], cost: 1
20.75/16.43	
20.75/16.43	      4: start -> eval : A'=1+A, B'=A+B, [ A>=B ], cost: 2
20.75/16.43	
20.75/16.43	      5: start -> eval : A'=-1+1/2*k+A-k*C+1/2*k^2, B'=1-k^2*C-11/6*k+2*C+k*C^2+1/3*k^3-k*C+1/2*k^2+B, C'=-k+C, [ A>=B && k>0 && -3/2-(-1+k)*C+1/2*k+1/2*(-1+k)^2+A>=17/6-(-1+k)*C-11/6*k+2*C+(-1+k)*C^2+1/2*(-1+k)^2+1/3*(-1+k)^3-(-1+k)^2*C+B ], cost: 1+k
20.75/16.43	
20.75/16.43	
20.75/16.43	
20.75/16.43	Removed unreachable locations (and leaf rules with constant cost):
20.75/16.43	
20.75/16.43	   Start location: start
20.75/16.43	
20.75/16.43	      5: start -> eval : A'=-1+1/2*k+A-k*C+1/2*k^2, B'=1-k^2*C-11/6*k+2*C+k*C^2+1/3*k^3-k*C+1/2*k^2+B, C'=-k+C, [ A>=B && k>0 && -3/2-(-1+k)*C+1/2*k+1/2*(-1+k)^2+A>=17/6-(-1+k)*C-11/6*k+2*C+(-1+k)*C^2+1/2*(-1+k)^2+1/3*(-1+k)^3-(-1+k)^2*C+B ], cost: 1+k
20.75/16.43	
20.75/16.43	
20.75/16.43	
20.75/16.43	### Computing asymptotic complexity ###
20.75/16.43	
20.75/16.43	
20.75/16.43	
20.75/16.43	Fully simplified ITS problem
20.75/16.43	
20.75/16.43	   Start location: start
20.75/16.43	
20.75/16.43	      5: start -> eval : A'=-1+1/2*k+A-k*C+1/2*k^2, B'=1-k^2*C-11/6*k+2*C+k*C^2+1/3*k^3-k*C+1/2*k^2+B, C'=-k+C, [ A>=B && k>0 && -3/2-(-1+k)*C+1/2*k+1/2*(-1+k)^2+A>=17/6-(-1+k)*C-11/6*k+2*C+(-1+k)*C^2+1/2*(-1+k)^2+1/3*(-1+k)^3-(-1+k)^2*C+B ], cost: 1+k
20.75/16.43	
20.75/16.43	
20.75/16.43	
20.75/16.43	Computing asymptotic complexity for rule 5
20.75/16.43	
20.75/16.43	   Could not solve the limit problem.
20.75/16.43	
20.75/16.43	   Resulting cost 0 has complexity: Unknown
20.75/16.43	
20.75/16.43	
20.75/16.43	
20.75/16.43	Obtained the following overall complexity (w.r.t. the length of the input n):
20.75/16.43	
20.75/16.43	   Complexity:  Unknown
20.75/16.43	
20.75/16.43	   Cpx degree:  ?
20.75/16.43	
20.75/16.43	   Solved cost: 0
20.75/16.43	
20.75/16.43	   Rule cost:   0
20.75/16.43	
20.75/16.43	   Rule guard:  []
20.75/16.43	
20.75/16.43	
20.75/16.43	
20.75/16.43	WORST_CASE(Omega(0),?)
20.75/16.43	
20.75/16.43	
20.75/16.43	----------------------------------------
20.75/16.43	
20.75/16.43	(2)
20.75/16.43	BOUNDS(1, INF)
20.83/16.69	EOF