3.91/1.91	WORST_CASE(Omega(n^1), O(n^1))
3.91/1.92	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.91/1.92	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.91/1.92	
3.91/1.92	
3.91/1.92	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(Arg_1 + nat(Arg_0), Arg_1, 0) + max(2, 2 + Arg_0)).
3.91/1.92	
3.91/1.92	(0) CpxIntTrs
3.91/1.92	(1) Koat2 Proof [FINISHED, 226 ms]
3.91/1.92	(2) BOUNDS(1, max(Arg_1 + nat(Arg_0), Arg_1, 0) + max(2, 2 + Arg_0))
3.91/1.92	(3) Loat Proof [FINISHED, 322 ms]
3.91/1.92	(4) BOUNDS(n^1, INF)
3.91/1.92	
3.91/1.92	
3.91/1.92	----------------------------------------
3.91/1.92	
3.91/1.92	(0)
3.91/1.92	Obligation:
3.91/1.92	Complexity Int TRS consisting of the following rules:
3.91/1.92	l0(A, B) -> Com_1(l1(A, B)) :|: TRUE
3.91/1.92	l1(A, B) -> Com_1(l1(A - 1, B + 1)) :|: A >= 1
3.91/1.92	l1(A, B) -> Com_1(l2(A, B)) :|: 0 >= A
3.91/1.92	l2(A, B) -> Com_1(l2(A, B - 1)) :|: B >= 1
3.91/1.92	
3.91/1.92	The start-symbols are:[l0_2]
3.91/1.92	
3.91/1.92	
3.91/1.92	----------------------------------------
3.91/1.92	
3.91/1.92	(1) Koat2 Proof (FINISHED)
3.91/1.92	YES( ?, max([0, max([Arg_1, Arg_1+max([0, Arg_0])])])+max([2, 2+Arg_0]) {O(n)})
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Initial Complexity Problem:
3.91/1.92	
3.91/1.92	  Start:  l0  
3.91/1.92	
3.91/1.92	  Program_Vars:  Arg_0, Arg_1  
3.91/1.92	
3.91/1.92	  Temp_Vars:    
3.91/1.92	
3.91/1.92	  Locations:  l0, l1, l2  
3.91/1.92	
3.91/1.92	  Transitions:
3.91/1.92	
3.91/1.92	  l0(Arg_0,Arg_1) -> l1(Arg_0,Arg_1):|:
3.91/1.92	
3.91/1.92	  l1(Arg_0,Arg_1) -> l1(Arg_0-1,Arg_1+1):|:1 <= Arg_0
3.91/1.92	
3.91/1.92	  l1(Arg_0,Arg_1) -> l2(Arg_0,Arg_1):|:Arg_0 <= 0
3.91/1.92	
3.91/1.92	  l2(Arg_0,Arg_1) -> l2(Arg_0,Arg_1-1):|:Arg_0 <= 0 && 1 <= Arg_1  
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Timebounds: 
3.91/1.92	
3.91/1.92	  Overall timebound: max([0, max([Arg_1, Arg_1+max([0, Arg_0])])])+max([2, 2+Arg_0]) {O(n)}
3.91/1.92	
3.91/1.92	  0: l0->l1: 1 {O(1)}
3.91/1.92	
3.91/1.92	  1: l1->l1: max([0, Arg_0]) {O(n)}
3.91/1.92	
3.91/1.92	  2: l1->l2: 1 {O(1)}
3.91/1.92	
3.91/1.92	  3: l2->l2: max([0, max([Arg_1, Arg_1+max([0, Arg_0])])]) {O(n)}
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Costbounds:
3.91/1.92	
3.91/1.92	  Overall costbound: max([0, max([Arg_1, Arg_1+max([0, Arg_0])])])+max([2, 2+Arg_0]) {O(n)}
3.91/1.92	
3.91/1.92	  0: l0->l1: 1 {O(1)}
3.91/1.92	
3.91/1.92	  1: l1->l1: max([0, Arg_0]) {O(n)}
3.91/1.92	
3.91/1.92	  2: l1->l2: 1 {O(1)}
3.91/1.92	
3.91/1.92	  3: l2->l2: max([0, max([Arg_1, Arg_1+max([0, Arg_0])])]) {O(n)}
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Sizebounds:
3.91/1.92	
3.91/1.92	`Lower:
3.91/1.92	
3.91/1.92	  0: l0->l1, Arg_0: Arg_0 {O(n)}
3.91/1.92	
3.91/1.92	  0: l0->l1, Arg_1: Arg_1 {O(n)}
3.91/1.92	
3.91/1.92	  1: l1->l1, Arg_0: 0 {O(1)}
3.91/1.92	
3.91/1.92	  1: l1->l1, Arg_1: Arg_1 {O(n)}
3.91/1.92	
3.91/1.92	  2: l1->l2, Arg_0: min([0, Arg_0]) {O(n)}
3.91/1.92	
3.91/1.92	  2: l1->l2, Arg_1: Arg_1 {O(n)}
3.91/1.92	
3.91/1.92	  3: l2->l2, Arg_0: min([0, Arg_0]) {O(n)}
3.91/1.92	
3.91/1.92	  3: l2->l2, Arg_1: 0 {O(1)}
3.91/1.92	
3.91/1.92	`Upper:
3.91/1.92	
3.91/1.92	  0: l0->l1, Arg_0: Arg_0 {O(n)}
3.91/1.92	
3.91/1.92	  0: l0->l1, Arg_1: Arg_1 {O(n)}
3.91/1.92	
3.91/1.92	  1: l1->l1, Arg_0: Arg_0 {O(n)}
3.91/1.92	
3.91/1.92	  1: l1->l1, Arg_1: Arg_1+max([0, Arg_0]) {O(n)}
3.91/1.92	
3.91/1.92	  2: l1->l2, Arg_0: 0 {O(1)}
3.91/1.92	
3.91/1.92	  2: l1->l2, Arg_1: max([Arg_1, Arg_1+max([0, Arg_0])]) {O(n)}
3.91/1.92	
3.91/1.92	  3: l2->l2, Arg_0: 0 {O(1)}
3.91/1.92	
3.91/1.92	  3: l2->l2, Arg_1: max([Arg_1, Arg_1+max([0, Arg_0])]) {O(n)}
3.91/1.92	
3.91/1.92	
3.91/1.92	----------------------------------------
3.91/1.92	
3.91/1.92	(2)
3.91/1.92	BOUNDS(1, max(Arg_1 + nat(Arg_0), Arg_1, 0) + max(2, 2 + Arg_0))
3.91/1.92	
3.91/1.92	----------------------------------------
3.91/1.92	
3.91/1.92	(3) Loat Proof (FINISHED)
3.91/1.92	
3.91/1.92	
3.91/1.92	### Pre-processing the ITS problem ###
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Initial linear ITS problem
3.91/1.92	
3.91/1.92	   Start location: l0
3.91/1.92	
3.91/1.92	      0: l0 -> l1 : [], cost: 1
3.91/1.92	
3.91/1.92	      1: l1 -> l1 : A'=-1+A, B'=1+B, [ A>=1 ], cost: 1
3.91/1.92	
3.91/1.92	      2: l1 -> l2 : [ 0>=A ], cost: 1
3.91/1.92	
3.91/1.92	      3: l2 -> l2 : B'=-1+B, [ B>=1 ], cost: 1
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	### Simplification by acceleration and chaining ###
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Accelerating simple loops of location 1.
3.91/1.92	
3.91/1.92	   Accelerating the following rules:
3.91/1.92	
3.91/1.92	      1: l1 -> l1 : A'=-1+A, B'=1+B, [ A>=1 ], cost: 1
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	   Accelerated rule 1 with metering function A, yielding the new rule 4.
3.91/1.92	
3.91/1.92	   Removing the simple loops: 1.
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Accelerating simple loops of location 2.
3.91/1.92	
3.91/1.92	   Accelerating the following rules:
3.91/1.92	
3.91/1.92	      3: l2 -> l2 : B'=-1+B, [ B>=1 ], cost: 1
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	   Accelerated rule 3 with metering function B, yielding the new rule 5.
3.91/1.92	
3.91/1.92	   Removing the simple loops: 3.
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Accelerated all simple loops using metering functions (where possible):
3.91/1.92	
3.91/1.92	   Start location: l0
3.91/1.92	
3.91/1.92	      0: l0 -> l1 : [], cost: 1
3.91/1.92	
3.91/1.92	      2: l1 -> l2 : [ 0>=A ], cost: 1
3.91/1.92	
3.91/1.92	      4: l1 -> l1 : A'=0, B'=A+B, [ A>=1 ], cost: A
3.91/1.92	
3.91/1.92	      5: l2 -> l2 : B'=0, [ B>=1 ], cost: B
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Chained accelerated rules (with incoming rules):
3.91/1.92	
3.91/1.92	   Start location: l0
3.91/1.92	
3.91/1.92	      0: l0 -> l1 : [], cost: 1
3.91/1.92	
3.91/1.92	      6: l0 -> l1 : A'=0, B'=A+B, [ A>=1 ], cost: 1+A
3.91/1.92	
3.91/1.92	      2: l1 -> l2 : [ 0>=A ], cost: 1
3.91/1.92	
3.91/1.92	      7: l1 -> l2 : B'=0, [ 0>=A && B>=1 ], cost: 1+B
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Removed unreachable locations (and leaf rules with constant cost):
3.91/1.92	
3.91/1.92	   Start location: l0
3.91/1.92	
3.91/1.92	      0: l0 -> l1 : [], cost: 1
3.91/1.92	
3.91/1.92	      6: l0 -> l1 : A'=0, B'=A+B, [ A>=1 ], cost: 1+A
3.91/1.92	
3.91/1.92	      7: l1 -> l2 : B'=0, [ 0>=A && B>=1 ], cost: 1+B
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Eliminated locations (on tree-shaped paths):
3.91/1.92	
3.91/1.92	   Start location: l0
3.91/1.92	
3.91/1.92	      8: l0 -> l2 : B'=0, [ 0>=A && B>=1 ], cost: 2+B
3.91/1.92	
3.91/1.92	      9: l0 -> l2 : A'=0, B'=0, [ A>=1 && A+B>=1 ], cost: 2+2*A+B
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	### Computing asymptotic complexity ###
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Fully simplified ITS problem
3.91/1.92	
3.91/1.92	   Start location: l0
3.91/1.92	
3.91/1.92	      8: l0 -> l2 : B'=0, [ 0>=A && B>=1 ], cost: 2+B
3.91/1.92	
3.91/1.92	      9: l0 -> l2 : A'=0, B'=0, [ A>=1 && A+B>=1 ], cost: 2+2*A+B
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Computing asymptotic complexity for rule 8
3.91/1.92	
3.91/1.92	   Solved the limit problem by the following transformations:
3.91/1.92	
3.91/1.92	      Created initial limit problem:
3.91/1.92	
3.91/1.92	      2+B (+), 1-A (+/+!), B (+/+!) [not solved]
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	      removing all constraints (solved by SMT)
3.91/1.92	
3.91/1.92	      resulting limit problem: [solved]
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	      applying transformation rule (C) using substitution {A==-n,B==n}
3.91/1.92	
3.91/1.92	      resulting limit problem:
3.91/1.92	
3.91/1.92	      [solved]
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	   Solution:
3.91/1.92	
3.91/1.92	      A / -n
3.91/1.92	
3.91/1.92	      B / n
3.91/1.92	
3.91/1.92	   Resulting cost 2+n has complexity: Poly(n^1)
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Found new complexity Poly(n^1).
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	Obtained the following overall complexity (w.r.t. the length of the input n):
3.91/1.92	
3.91/1.92	   Complexity:  Poly(n^1)
3.91/1.92	
3.91/1.92	   Cpx degree:  1
3.91/1.92	
3.91/1.92	   Solved cost: 2+n
3.91/1.92	
3.91/1.92	   Rule cost:   2+B
3.91/1.92	
3.91/1.92	   Rule guard:  [ 0>=A && B>=1 ]
3.91/1.92	
3.91/1.92	
3.91/1.92	
3.91/1.92	WORST_CASE(Omega(n^1),?)
3.91/1.92	
3.91/1.92	
3.91/1.92	----------------------------------------
3.91/1.92	
3.91/1.92	(4)
3.91/1.92	BOUNDS(n^1, INF)
4.02/1.94	EOF