3.90/1.99	WORST_CASE(Omega(n^1), O(n^1))
3.90/1.99	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.90/1.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.90/1.99	
3.90/1.99	
3.90/1.99	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_1) + nat(Arg_0)).
3.90/1.99	
3.90/1.99	(0) CpxIntTrs
3.90/1.99	(1) Koat2 Proof [FINISHED, 133 ms]
3.90/1.99	(2) BOUNDS(1, max(1, 1 + Arg_1) + nat(Arg_0))
3.90/1.99	(3) Loat Proof [FINISHED, 316 ms]
3.90/1.99	(4) BOUNDS(n^1, INF)
3.90/1.99	
3.90/1.99	
3.90/1.99	----------------------------------------
3.90/1.99	
3.90/1.99	(0)
3.90/1.99	Obligation:
3.90/1.99	Complexity Int TRS consisting of the following rules:
3.90/1.99	merge(A, B) -> Com_1(merge(A - 1, B)) :|: A >= 1 && B >= 1
3.90/1.99	merge(A, B) -> Com_1(merge(A, B - 1)) :|: A >= 1 && B >= 1
3.90/1.99	start(A, B) -> Com_1(merge(A, B)) :|: TRUE
3.90/1.99	
3.90/1.99	The start-symbols are:[start_2]
3.90/1.99	
3.90/1.99	
3.90/1.99	----------------------------------------
3.90/1.99	
3.90/1.99	(1) Koat2 Proof (FINISHED)
3.90/1.99	YES( ?, 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)})
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	Initial Complexity Problem:
3.90/1.99	
3.90/1.99	  Start:  start  
3.90/1.99	
3.90/1.99	  Program_Vars:  Arg_0, Arg_1  
3.90/1.99	
3.90/1.99	  Temp_Vars:    
3.90/1.99	
3.90/1.99	  Locations:  merge, start  
3.90/1.99	
3.90/1.99	  Transitions:
3.90/1.99	
3.90/1.99	  merge(Arg_0,Arg_1) -> merge(Arg_0-1,Arg_1):|:1 <= Arg_0 && 1 <= Arg_1
3.90/1.99	
3.90/1.99	  merge(Arg_0,Arg_1) -> merge(Arg_0,Arg_1-1):|:1 <= Arg_0 && 1 <= Arg_1
3.90/1.99	
3.90/1.99	  start(Arg_0,Arg_1) -> merge(Arg_0,Arg_1):|:  
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	Timebounds: 
3.90/1.99	
3.90/1.99	  Overall timebound: 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)}
3.90/1.99	
3.90/1.99	  0: merge->merge: max([0, Arg_0]) {O(n)}
3.90/1.99	
3.90/1.99	  1: merge->merge: max([0, Arg_1]) {O(n)}
3.90/1.99	
3.90/1.99	  2: start->merge: 1 {O(1)}
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	Costbounds:
3.90/1.99	
3.90/1.99	  Overall costbound: 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)}
3.90/1.99	
3.90/1.99	  0: merge->merge: max([0, Arg_0]) {O(n)}
3.90/1.99	
3.90/1.99	  1: merge->merge: max([0, Arg_1]) {O(n)}
3.90/1.99	
3.90/1.99	  2: start->merge: 1 {O(1)}
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	Sizebounds:
3.90/1.99	
3.90/1.99	`Lower:
3.90/1.99	
3.90/1.99	  0: merge->merge, Arg_0: 0 {O(1)}
3.90/1.99	
3.90/1.99	  0: merge->merge, Arg_1: 1 {O(1)}
3.90/1.99	
3.90/1.99	  1: merge->merge, Arg_0: 1 {O(1)}
3.90/1.99	
3.90/1.99	  1: merge->merge, Arg_1: 0 {O(1)}
3.90/1.99	
3.90/1.99	  2: start->merge, Arg_0: Arg_0 {O(n)}
3.90/1.99	
3.90/1.99	  2: start->merge, Arg_1: Arg_1 {O(n)}
3.90/1.99	
3.90/1.99	`Upper:
3.90/1.99	
3.90/1.99	  0: merge->merge, Arg_0: Arg_0 {O(n)}
3.90/1.99	
3.90/1.99	  0: merge->merge, Arg_1: Arg_1 {O(n)}
3.90/1.99	
3.90/1.99	  1: merge->merge, Arg_0: Arg_0 {O(n)}
3.90/1.99	
3.90/1.99	  1: merge->merge, Arg_1: Arg_1 {O(n)}
3.90/1.99	
3.90/1.99	  2: start->merge, Arg_0: Arg_0 {O(n)}
3.90/1.99	
3.90/1.99	  2: start->merge, Arg_1: Arg_1 {O(n)}
3.90/1.99	
3.90/1.99	
3.90/1.99	----------------------------------------
3.90/1.99	
3.90/1.99	(2)
3.90/1.99	BOUNDS(1, max(1, 1 + Arg_1) + nat(Arg_0))
3.90/1.99	
3.90/1.99	----------------------------------------
3.90/1.99	
3.90/1.99	(3) Loat Proof (FINISHED)
3.90/1.99	
3.90/1.99	
3.90/1.99	### Pre-processing the ITS problem ###
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	Initial linear ITS problem
3.90/1.99	
3.90/1.99	   Start location: start
3.90/1.99	
3.90/1.99	      0: merge -> merge : A'=-1+A, [ A>=1 && B>=1 ], cost: 1
3.90/1.99	
3.90/1.99	      1: merge -> merge : B'=-1+B, [ A>=1 && B>=1 ], cost: 1
3.90/1.99	
3.90/1.99	      2: start -> merge : [], cost: 1
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	### Simplification by acceleration and chaining ###
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	Accelerating simple loops of location 0.
3.90/1.99	
3.90/1.99	   Accelerating the following rules:
3.90/1.99	
3.90/1.99	      0: merge -> merge : A'=-1+A, [ A>=1 && B>=1 ], cost: 1
3.90/1.99	
3.90/1.99	      1: merge -> merge : B'=-1+B, [ A>=1 && B>=1 ], cost: 1
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	   Accelerated rule 0 with metering function A, yielding the new rule 3.
3.90/1.99	
3.90/1.99	   Accelerated rule 1 with metering function B, yielding the new rule 4.
3.90/1.99	
3.90/1.99	   Removing the simple loops: 0 1.
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	Accelerated all simple loops using metering functions (where possible):
3.90/1.99	
3.90/1.99	   Start location: start
3.90/1.99	
3.90/1.99	      3: merge -> merge : A'=0, [ A>=1 && B>=1 ], cost: A
3.90/1.99	
3.90/1.99	      4: merge -> merge : B'=0, [ A>=1 && B>=1 ], cost: B
3.90/1.99	
3.90/1.99	      2: start -> merge : [], cost: 1
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	Chained accelerated rules (with incoming rules):
3.90/1.99	
3.90/1.99	   Start location: start
3.90/1.99	
3.90/1.99	      2: start -> merge : [], cost: 1
3.90/1.99	
3.90/1.99	      5: start -> merge : A'=0, [ A>=1 && B>=1 ], cost: 1+A
3.90/1.99	
3.90/1.99	      6: start -> merge : B'=0, [ A>=1 && B>=1 ], cost: 1+B
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	Removed unreachable locations (and leaf rules with constant cost):
3.90/1.99	
3.90/1.99	   Start location: start
3.90/1.99	
3.90/1.99	      5: start -> merge : A'=0, [ A>=1 && B>=1 ], cost: 1+A
3.90/1.99	
3.90/1.99	      6: start -> merge : B'=0, [ A>=1 && B>=1 ], cost: 1+B
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	### Computing asymptotic complexity ###
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	Fully simplified ITS problem
3.90/1.99	
3.90/1.99	   Start location: start
3.90/1.99	
3.90/1.99	      5: start -> merge : A'=0, [ A>=1 && B>=1 ], cost: 1+A
3.90/1.99	
3.90/1.99	      6: start -> merge : B'=0, [ A>=1 && B>=1 ], cost: 1+B
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	Computing asymptotic complexity for rule 5
3.90/1.99	
3.90/1.99	   Solved the limit problem by the following transformations:
3.90/1.99	
3.90/1.99	      Created initial limit problem:
3.90/1.99	
3.90/1.99	      A (+/+!), B (+/+!), 1+A (+) [not solved]
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	      removing all constraints (solved by SMT)
3.90/1.99	
3.90/1.99	      resulting limit problem: [solved]
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	      applying transformation rule (C) using substitution {A==n,B==1}
3.90/1.99	
3.90/1.99	      resulting limit problem:
3.90/1.99	
3.90/1.99	      [solved]
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	   Solution:
3.90/1.99	
3.90/1.99	      A / n
3.90/1.99	
3.90/1.99	      B / 1
3.90/1.99	
3.90/1.99	   Resulting cost 1+n has complexity: Poly(n^1)
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	Found new complexity Poly(n^1).
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	Obtained the following overall complexity (w.r.t. the length of the input n):
3.90/1.99	
3.90/1.99	   Complexity:  Poly(n^1)
3.90/1.99	
3.90/1.99	   Cpx degree:  1
3.90/1.99	
3.90/1.99	   Solved cost: 1+n
3.90/1.99	
3.90/1.99	   Rule cost:   1+A
3.90/1.99	
3.90/1.99	   Rule guard:  [ A>=1 && B>=1 ]
3.90/1.99	
3.90/1.99	
3.90/1.99	
3.90/1.99	WORST_CASE(Omega(n^1),?)
3.90/1.99	
3.90/1.99	
3.90/1.99	----------------------------------------
3.90/1.99	
3.90/1.99	(4)
3.90/1.99	BOUNDS(n^1, INF)
3.90/2.02	EOF