3.63/1.98	WORST_CASE(Omega(n^1), O(n^1))
3.63/1.99	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
3.63/1.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.63/1.99	
3.63/1.99	
3.63/1.99	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + -1 * Arg_2) + nat(Arg_0 + -1 * Arg_1)).
3.63/1.99	
3.63/1.99	(0) CpxIntTrs
3.63/1.99	(1) Koat2 Proof [FINISHED, 160 ms]
3.63/1.99	(2) BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_2) + nat(Arg_0 + -1 * Arg_1))
3.63/1.99	(3) Loat Proof [FINISHED, 342 ms]
3.63/1.99	(4) BOUNDS(n^1, INF)
3.63/1.99	
3.63/1.99	
3.63/1.99	----------------------------------------
3.63/1.99	
3.63/1.99	(0)
3.63/1.99	Obligation:
3.63/1.99	Complexity Int TRS consisting of the following rules:
3.63/1.99	f(A, B, C) -> Com_1(f(A, B + 1, C)) :|: A >= B + 1 && A >= C + 1
3.63/1.99	f(A, B, C) -> Com_1(f(A, B, C + 1)) :|: A >= B + 1 && A >= C + 1
3.63/1.99	start(A, B, C) -> Com_1(f(A, B, C)) :|: TRUE
3.63/1.99	
3.63/1.99	The start-symbols are:[start_3]
3.63/1.99	
3.63/1.99	
3.63/1.99	----------------------------------------
3.63/1.99	
3.63/1.99	(1) Koat2 Proof (FINISHED)
3.63/1.99	YES( ?, 1+max([0, Arg_0-Arg_2])+max([0, Arg_0-Arg_1]) {O(n)})
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	Initial Complexity Problem:
3.63/1.99	
3.63/1.99	  Start:  start  
3.63/1.99	
3.63/1.99	  Program_Vars:  Arg_0, Arg_1, Arg_2  
3.63/1.99	
3.63/1.99	  Temp_Vars:    
3.63/1.99	
3.63/1.99	  Locations:  f, start  
3.63/1.99	
3.63/1.99	  Transitions:
3.63/1.99	
3.63/1.99	  f(Arg_0,Arg_1,Arg_2) -> f(Arg_0,Arg_1+1,Arg_2):|:Arg_1+1 <= Arg_0 && Arg_2+1 <= Arg_0
3.63/1.99	
3.63/1.99	  f(Arg_0,Arg_1,Arg_2) -> f(Arg_0,Arg_1,Arg_2+1):|:Arg_1+1 <= Arg_0 && Arg_2+1 <= Arg_0
3.63/1.99	
3.63/1.99	  start(Arg_0,Arg_1,Arg_2) -> f(Arg_0,Arg_1,Arg_2):|:  
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	Timebounds: 
3.63/1.99	
3.63/1.99	  Overall timebound: 1+max([0, Arg_0-Arg_2])+max([0, Arg_0-Arg_1]) {O(n)}
3.63/1.99	
3.63/1.99	  0: f->f: max([0, Arg_0-Arg_1]) {O(n)}
3.63/1.99	
3.63/1.99	  1: f->f: max([0, Arg_0-Arg_2]) {O(n)}
3.63/1.99	
3.63/1.99	  2: start->f: 1 {O(1)}
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	Costbounds:
3.63/1.99	
3.63/1.99	  Overall costbound: 1+max([0, Arg_0-Arg_2])+max([0, Arg_0-Arg_1]) {O(n)}
3.63/1.99	
3.63/1.99	  0: f->f: max([0, Arg_0-Arg_1]) {O(n)}
3.63/1.99	
3.63/1.99	  1: f->f: max([0, Arg_0-Arg_2]) {O(n)}
3.63/1.99	
3.63/1.99	  2: start->f: 1 {O(1)}
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	Sizebounds:
3.63/1.99	
3.63/1.99	`Lower:
3.63/1.99	
3.63/1.99	  0: f->f, Arg_0: Arg_0 {O(n)}
3.63/1.99	
3.63/1.99	  0: f->f, Arg_1: Arg_1 {O(n)}
3.63/1.99	
3.63/1.99	  0: f->f, Arg_2: Arg_2 {O(n)}
3.63/1.99	
3.63/1.99	  1: f->f, Arg_0: Arg_0 {O(n)}
3.63/1.99	
3.63/1.99	  1: f->f, Arg_1: Arg_1 {O(n)}
3.63/1.99	
3.63/1.99	  1: f->f, Arg_2: Arg_2 {O(n)}
3.63/1.99	
3.63/1.99	  2: start->f, Arg_0: Arg_0 {O(n)}
3.63/1.99	
3.63/1.99	  2: start->f, Arg_1: Arg_1 {O(n)}
3.63/1.99	
3.63/1.99	  2: start->f, Arg_2: Arg_2 {O(n)}
3.63/1.99	
3.63/1.99	`Upper:
3.63/1.99	
3.63/1.99	  0: f->f, Arg_0: Arg_0 {O(n)}
3.63/1.99	
3.63/1.99	  0: f->f, Arg_1: Arg_1+max([0, Arg_0-Arg_1]) {O(n)}
3.63/1.99	
3.63/1.99	  0: f->f, Arg_2: Arg_2+max([0, Arg_0-Arg_2]) {O(n)}
3.63/1.99	
3.63/1.99	  1: f->f, Arg_0: Arg_0 {O(n)}
3.63/1.99	
3.63/1.99	  1: f->f, Arg_1: Arg_1+max([0, Arg_0-Arg_1]) {O(n)}
3.63/1.99	
3.63/1.99	  1: f->f, Arg_2: Arg_2+max([0, Arg_0-Arg_2]) {O(n)}
3.63/1.99	
3.63/1.99	  2: start->f, Arg_0: Arg_0 {O(n)}
3.63/1.99	
3.63/1.99	  2: start->f, Arg_1: Arg_1 {O(n)}
3.63/1.99	
3.63/1.99	  2: start->f, Arg_2: Arg_2 {O(n)}
3.63/1.99	
3.63/1.99	
3.63/1.99	----------------------------------------
3.63/1.99	
3.63/1.99	(2)
3.63/1.99	BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_2) + nat(Arg_0 + -1 * Arg_1))
3.63/1.99	
3.63/1.99	----------------------------------------
3.63/1.99	
3.63/1.99	(3) Loat Proof (FINISHED)
3.63/1.99	
3.63/1.99	
3.63/1.99	### Pre-processing the ITS problem ###
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	Initial linear ITS problem
3.63/1.99	
3.63/1.99	   Start location: start
3.63/1.99	
3.63/1.99	      0: f -> f : B'=1+B, [ A>=1+B && A>=1+C ], cost: 1
3.63/1.99	
3.63/1.99	      1: f -> f : C'=1+C, [ A>=1+B && A>=1+C ], cost: 1
3.63/1.99	
3.63/1.99	      2: start -> f : [], cost: 1
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	### Simplification by acceleration and chaining ###
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	Accelerating simple loops of location 0.
3.63/1.99	
3.63/1.99	   Accelerating the following rules:
3.63/1.99	
3.63/1.99	      0: f -> f : B'=1+B, [ A>=1+B && A>=1+C ], cost: 1
3.63/1.99	
3.63/1.99	      1: f -> f : C'=1+C, [ A>=1+B && A>=1+C ], cost: 1
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	   Accelerated rule 0 with metering function A-B, yielding the new rule 3.
3.63/1.99	
3.63/1.99	   Accelerated rule 1 with metering function -C+A, yielding the new rule 4.
3.63/1.99	
3.63/1.99	   Removing the simple loops: 0 1.
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	Accelerated all simple loops using metering functions (where possible):
3.63/1.99	
3.63/1.99	   Start location: start
3.63/1.99	
3.63/1.99	      3: f -> f : B'=A, [ A>=1+B && A>=1+C ], cost: A-B
3.63/1.99	
3.63/1.99	      4: f -> f : C'=A, [ A>=1+B && A>=1+C ], cost: -C+A
3.63/1.99	
3.63/1.99	      2: start -> f : [], cost: 1
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	Chained accelerated rules (with incoming rules):
3.63/1.99	
3.63/1.99	   Start location: start
3.63/1.99	
3.63/1.99	      2: start -> f : [], cost: 1
3.63/1.99	
3.63/1.99	      5: start -> f : B'=A, [ A>=1+B && A>=1+C ], cost: 1+A-B
3.63/1.99	
3.63/1.99	      6: start -> f : C'=A, [ A>=1+B && A>=1+C ], cost: 1-C+A
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	Removed unreachable locations (and leaf rules with constant cost):
3.63/1.99	
3.63/1.99	   Start location: start
3.63/1.99	
3.63/1.99	      5: start -> f : B'=A, [ A>=1+B && A>=1+C ], cost: 1+A-B
3.63/1.99	
3.63/1.99	      6: start -> f : C'=A, [ A>=1+B && A>=1+C ], cost: 1-C+A
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	### Computing asymptotic complexity ###
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	Fully simplified ITS problem
3.63/1.99	
3.63/1.99	   Start location: start
3.63/1.99	
3.63/1.99	      5: start -> f : B'=A, [ A>=1+B && A>=1+C ], cost: 1+A-B
3.63/1.99	
3.63/1.99	      6: start -> f : C'=A, [ A>=1+B && A>=1+C ], cost: 1-C+A
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	Computing asymptotic complexity for rule 5
3.63/1.99	
3.63/1.99	   Solved the limit problem by the following transformations:
3.63/1.99	
3.63/1.99	      Created initial limit problem:
3.63/1.99	
3.63/1.99	      -C+A (+/+!), 1+A-B (+), A-B (+/+!) [not solved]
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	      removing all constraints (solved by SMT)
3.63/1.99	
3.63/1.99	      resulting limit problem: [solved]
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	      applying transformation rule (C) using substitution {C==0,A==1,B==-n}
3.63/1.99	
3.63/1.99	      resulting limit problem:
3.63/1.99	
3.63/1.99	      [solved]
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	   Solution:
3.63/1.99	
3.63/1.99	      C / 0
3.63/1.99	
3.63/1.99	      A / 1
3.63/1.99	
3.63/1.99	      B / -n
3.63/1.99	
3.63/1.99	   Resulting cost 2+n has complexity: Poly(n^1)
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	Found new complexity Poly(n^1).
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	Obtained the following overall complexity (w.r.t. the length of the input n):
3.63/1.99	
3.63/1.99	   Complexity:  Poly(n^1)
3.63/1.99	
3.63/1.99	   Cpx degree:  1
3.63/1.99	
3.63/1.99	   Solved cost: 2+n
3.63/1.99	
3.63/1.99	   Rule cost:   1+A-B
3.63/1.99	
3.63/1.99	   Rule guard:  [ A>=1+B && A>=1+C ]
3.63/1.99	
3.63/1.99	
3.63/1.99	
3.63/1.99	WORST_CASE(Omega(n^1),?)
3.63/1.99	
3.63/1.99	
3.63/1.99	----------------------------------------
3.63/1.99	
3.63/1.99	(4)
3.63/1.99	BOUNDS(n^1, INF)
4.04/2.02	EOF