3.92/1.87	WORST_CASE(Omega(n^1), O(n^1))
3.92/1.88	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.92/1.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.92/1.88	
3.92/1.88	
3.92/1.88	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + -1 * Arg_1)).
3.92/1.88	
3.92/1.88	(0) CpxIntTrs
3.92/1.88	(1) Koat2 Proof [FINISHED, 106 ms]
3.92/1.88	(2) BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
3.92/1.88	(3) Loat Proof [FINISHED, 211 ms]
3.92/1.88	(4) BOUNDS(n^1, INF)
3.92/1.88	
3.92/1.88	
3.92/1.88	----------------------------------------
3.92/1.88	
3.92/1.88	(0)
3.92/1.88	Obligation:
3.92/1.88	Complexity Int TRS consisting of the following rules:
3.92/1.88	eval(A, B, C) -> Com_1(eval(A - 1, B, C - 1)) :|: A >= B + 1 && C >= B + 1
3.92/1.88	start(A, B, C) -> Com_1(eval(A, B, C)) :|: TRUE
3.92/1.88	
3.92/1.88	The start-symbols are:[start_3]
3.92/1.88	
3.92/1.88	
3.92/1.88	----------------------------------------
3.92/1.88	
3.92/1.88	(1) Koat2 Proof (FINISHED)
3.92/1.88	YES( ?, max([1, 1+Arg_0-Arg_1]) {O(n)})
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	Initial Complexity Problem:
3.92/1.88	
3.92/1.88	  Start:  start  
3.92/1.88	
3.92/1.88	  Program_Vars:  Arg_0, Arg_1, Arg_2  
3.92/1.88	
3.92/1.88	  Temp_Vars:    
3.92/1.88	
3.92/1.88	  Locations:  eval, start  
3.92/1.88	
3.92/1.88	  Transitions:
3.92/1.88	
3.92/1.88	  eval(Arg_0,Arg_1,Arg_2) -> eval(Arg_0-1,Arg_1,Arg_2-1):|:Arg_1+1 <= Arg_0 && Arg_1+1 <= Arg_2
3.92/1.88	
3.92/1.88	  start(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1,Arg_2):|:  
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	Timebounds: 
3.92/1.88	
3.92/1.88	  Overall timebound: max([1, 1+Arg_0-Arg_1]) {O(n)}
3.92/1.88	
3.92/1.88	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
3.92/1.88	
3.92/1.88	  1: start->eval: 1 {O(1)}
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	Costbounds:
3.92/1.88	
3.92/1.88	  Overall costbound: max([1, 1+Arg_0-Arg_1]) {O(n)}
3.92/1.88	
3.92/1.88	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
3.92/1.88	
3.92/1.88	  1: start->eval: 1 {O(1)}
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	Sizebounds:
3.92/1.88	
3.92/1.88	`Lower:
3.92/1.88	
3.92/1.88	  0: eval->eval, Arg_0: Arg_0-max([0, Arg_0-Arg_1]) {O(n)}
3.92/1.88	
3.92/1.88	  0: eval->eval, Arg_1: Arg_1 {O(n)}
3.92/1.88	
3.92/1.88	  0: eval->eval, Arg_2: Arg_2-max([0, Arg_0-Arg_1]) {O(n)}
3.92/1.88	
3.92/1.88	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.92/1.88	
3.92/1.88	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.92/1.88	
3.92/1.88	  1: start->eval, Arg_2: Arg_2 {O(n)}
3.92/1.88	
3.92/1.88	`Upper:
3.92/1.88	
3.92/1.88	  0: eval->eval, Arg_0: Arg_0 {O(n)}
3.92/1.88	
3.92/1.88	  0: eval->eval, Arg_1: Arg_1 {O(n)}
3.92/1.88	
3.92/1.88	  0: eval->eval, Arg_2: Arg_2 {O(n)}
3.92/1.88	
3.92/1.88	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.92/1.88	
3.92/1.88	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.92/1.88	
3.92/1.88	  1: start->eval, Arg_2: Arg_2 {O(n)}
3.92/1.88	
3.92/1.88	
3.92/1.88	----------------------------------------
3.92/1.88	
3.92/1.88	(2)
3.92/1.88	BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
3.92/1.88	
3.92/1.88	----------------------------------------
3.92/1.88	
3.92/1.88	(3) Loat Proof (FINISHED)
3.92/1.88	
3.92/1.88	
3.92/1.88	### Pre-processing the ITS problem ###
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	Initial linear ITS problem
3.92/1.88	
3.92/1.88	   Start location: start
3.92/1.88	
3.92/1.88	      0: eval -> eval : A'=-1+A, C'=-1+C, [ A>=1+B && C>=1+B ], cost: 1
3.92/1.88	
3.92/1.88	      1: start -> eval : [], cost: 1
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	### Simplification by acceleration and chaining ###
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	Accelerating simple loops of location 0.
3.92/1.88	
3.92/1.88	   Accelerating the following rules:
3.92/1.88	
3.92/1.88	      0: eval -> eval : A'=-1+A, C'=-1+C, [ A>=1+B && C>=1+B ], cost: 1
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	   Accelerated rule 0 with metering function C-B (after adding A>=C), yielding the new rule 2.
3.92/1.88	
3.92/1.88	   Accelerated rule 0 with metering function A-B (after adding A<=C), yielding the new rule 3.
3.92/1.88	
3.92/1.88	   Removing the simple loops: 0.
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	Accelerated all simple loops using metering functions (where possible):
3.92/1.88	
3.92/1.88	   Start location: start
3.92/1.88	
3.92/1.88	      2: eval -> eval : A'=-C+A+B, C'=B, [ A>=1+B && C>=1+B && A>=C ], cost: C-B
3.92/1.88	
3.92/1.88	      3: eval -> eval : A'=B, C'=C-A+B, [ A>=1+B && C>=1+B && A<=C ], cost: A-B
3.92/1.88	
3.92/1.88	      1: start -> eval : [], cost: 1
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	Chained accelerated rules (with incoming rules):
3.92/1.88	
3.92/1.88	   Start location: start
3.92/1.88	
3.92/1.88	      1: start -> eval : [], cost: 1
3.92/1.88	
3.92/1.88	      4: start -> eval : A'=-C+A+B, C'=B, [ A>=1+B && C>=1+B && A>=C ], cost: 1+C-B
3.92/1.88	
3.92/1.88	      5: start -> eval : A'=B, C'=C-A+B, [ A>=1+B && C>=1+B && A<=C ], cost: 1+A-B
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	Removed unreachable locations (and leaf rules with constant cost):
3.92/1.88	
3.92/1.88	   Start location: start
3.92/1.88	
3.92/1.88	      4: start -> eval : A'=-C+A+B, C'=B, [ A>=1+B && C>=1+B && A>=C ], cost: 1+C-B
3.92/1.88	
3.92/1.88	      5: start -> eval : A'=B, C'=C-A+B, [ A>=1+B && C>=1+B && A<=C ], cost: 1+A-B
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	### Computing asymptotic complexity ###
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	Fully simplified ITS problem
3.92/1.88	
3.92/1.88	   Start location: start
3.92/1.88	
3.92/1.88	      4: start -> eval : A'=-C+A+B, C'=B, [ A>=1+B && C>=1+B && A>=C ], cost: 1+C-B
3.92/1.88	
3.92/1.88	      5: start -> eval : A'=B, C'=C-A+B, [ A>=1+B && C>=1+B && A<=C ], cost: 1+A-B
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	Computing asymptotic complexity for rule 4
3.92/1.88	
3.92/1.88	   Simplified the guard:
3.92/1.88	
3.92/1.88	      4: start -> eval : A'=-C+A+B, C'=B, [ C>=1+B && A>=C ], cost: 1+C-B
3.92/1.88	
3.92/1.88	   Solved the limit problem by the following transformations:
3.92/1.88	
3.92/1.88	      Created initial limit problem:
3.92/1.88	
3.92/1.88	      1+C-B (+), C-B (+/+!), 1-C+A (+/+!) [not solved]
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	      removing all constraints (solved by SMT)
3.92/1.88	
3.92/1.88	      resulting limit problem: [solved]
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	      applying transformation rule (C) using substitution {C==0,A==n,B==-n}
3.92/1.88	
3.92/1.88	      resulting limit problem:
3.92/1.88	
3.92/1.88	      [solved]
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	   Solution:
3.92/1.88	
3.92/1.88	      C / 0
3.92/1.88	
3.92/1.88	      A / n
3.92/1.88	
3.92/1.88	      B / -n
3.92/1.88	
3.92/1.88	   Resulting cost 1+n has complexity: Poly(n^1)
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	Found new complexity Poly(n^1).
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	Obtained the following overall complexity (w.r.t. the length of the input n):
3.92/1.88	
3.92/1.88	   Complexity:  Poly(n^1)
3.92/1.88	
3.92/1.88	   Cpx degree:  1
3.92/1.88	
3.92/1.88	   Solved cost: 1+n
3.92/1.88	
3.92/1.88	   Rule cost:   1+C-B
3.92/1.88	
3.92/1.88	   Rule guard:  [ C>=1+B && A>=C ]
3.92/1.88	
3.92/1.88	
3.92/1.88	
3.92/1.88	WORST_CASE(Omega(n^1),?)
3.92/1.88	
3.92/1.88	
3.92/1.88	----------------------------------------
3.92/1.88	
3.92/1.88	(4)
3.92/1.88	BOUNDS(n^1, INF)
3.94/2.47	EOF