3.63/1.99	WORST_CASE(Omega(n^1), O(n^1))
3.63/1.99	proof of /export/starexec/sandbox/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3.63/1.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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)).
3.63/1.99	
3.63/1.99	(0) CpxIntTrs
3.63/1.99	(1) Koat2 Proof [FINISHED, 28 ms]
3.63/1.99	(2) BOUNDS(1, max(1, 1 + Arg_0))
3.63/1.99	(3) Loat Proof [FINISHED, 251 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	eval(A, B) -> Com_1(eval(A - 1, B - 1)) :|: A >= 1 && B >= 1
3.63/1.99	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
3.63/1.99	
3.63/1.99	The start-symbols are:[start_2]
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( ?, max([1, 1+Arg_0]) {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  
3.63/1.99	
3.63/1.99	  Temp_Vars:    
3.63/1.99	
3.63/1.99	  Locations:  eval, start  
3.63/1.99	
3.63/1.99	  Transitions:
3.63/1.99	
3.63/1.99	  eval(Arg_0,Arg_1) -> eval(Arg_0-1,Arg_1-1):|:1 <= Arg_0 && 1 <= Arg_1
3.63/1.99	
3.63/1.99	  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  
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: max([1, 1+Arg_0]) {O(n)}
3.63/1.99	
3.63/1.99	  0: eval->eval: max([0, Arg_0]) {O(n)}
3.63/1.99	
3.63/1.99	  1: start->eval: 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: max([1, 1+Arg_0]) {O(n)}
3.63/1.99	
3.63/1.99	  0: eval->eval: max([0, Arg_0]) {O(n)}
3.63/1.99	
3.63/1.99	  1: start->eval: 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: eval->eval, Arg_0: 0 {O(1)}
3.63/1.99	
3.63/1.99	  0: eval->eval, Arg_1: 0 {O(1)}
3.63/1.99	
3.63/1.99	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.63/1.99	
3.63/1.99	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.63/1.99	
3.63/1.99	`Upper:
3.63/1.99	
3.63/1.99	  0: eval->eval, Arg_0: Arg_0 {O(n)}
3.63/1.99	
3.63/1.99	  0: eval->eval, Arg_1: Arg_1 {O(n)}
3.63/1.99	
3.63/1.99	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.63/1.99	
3.63/1.99	  1: start->eval, Arg_1: Arg_1 {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))
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: eval -> eval : A'=-1+A, B'=-1+B, [ A>=1 && B>=1 ], cost: 1
3.63/1.99	
3.63/1.99	      1: start -> eval : [], 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.
3.63/2.00	
3.63/2.00	   Accelerating the following rules:
3.63/2.00	
3.63/2.00	      0: eval -> eval : A'=-1+A, B'=-1+B, [ A>=1 && B>=1 ], cost: 1
3.63/2.00	
3.63/2.00	
3.63/2.00	
3.63/2.00	   Accelerated rule 0 with metering function B (after adding A>=B), yielding the new rule 2.
3.63/2.00	
3.63/2.00	   Accelerated rule 0 with metering function A (after adding A<=B), yielding the new rule 3.
3.63/2.00	
3.63/2.00	   Removing the simple loops: 0.
3.63/2.00	
3.63/2.00	
3.63/2.00	
3.63/2.00	Accelerated all simple loops using metering functions (where possible):
3.63/2.00	
3.63/2.00	   Start location: start
3.63/2.00	
3.63/2.00	      2: eval -> eval : A'=A-B, B'=0, [ A>=1 && B>=1 && A>=B ], cost: B
3.63/2.00	
3.63/2.00	      3: eval -> eval : A'=0, B'=-A+B, [ A>=1 && B>=1 && A<=B ], cost: A
3.63/2.00	
3.63/2.00	      1: start -> eval : [], cost: 1
3.63/2.00	
3.63/2.00	
3.63/2.00	
3.63/2.00	Chained accelerated rules (with incoming rules):
3.63/2.00	
3.63/2.00	   Start location: start
3.63/2.00	
3.63/2.00	      1: start -> eval : [], cost: 1
3.63/2.00	
3.63/2.00	      4: start -> eval : A'=A-B, B'=0, [ A>=1 && B>=1 && A>=B ], cost: 1+B
3.63/2.00	
3.63/2.00	      5: start -> eval : A'=0, B'=-A+B, [ A>=1 && B>=1 && A<=B ], cost: 1+A
3.63/2.00	
3.63/2.00	
3.63/2.00	
3.63/2.00	Removed unreachable locations (and leaf rules with constant cost):
3.63/2.00	
3.63/2.00	   Start location: start
3.63/2.00	
3.63/2.00	      4: start -> eval : A'=A-B, B'=0, [ A>=1 && B>=1 && A>=B ], cost: 1+B
3.63/2.00	
3.63/2.00	      5: start -> eval : A'=0, B'=-A+B, [ A>=1 && B>=1 && A<=B ], cost: 1+A
3.63/2.00	
3.63/2.00	
3.63/2.00	
3.63/2.00	### Computing asymptotic complexity ###
3.63/2.00	
3.63/2.00	
3.63/2.00	
3.63/2.00	Fully simplified ITS problem
3.63/2.00	
3.63/2.00	   Start location: start
3.63/2.00	
3.63/2.00	      4: start -> eval : A'=A-B, B'=0, [ A>=1 && B>=1 && A>=B ], cost: 1+B
3.63/2.00	
3.63/2.00	      5: start -> eval : A'=0, B'=-A+B, [ A>=1 && B>=1 && A<=B ], cost: 1+A
3.63/2.00	
3.63/2.00	
3.63/2.00	
3.63/2.00	Computing asymptotic complexity for rule 4
3.63/2.00	
3.63/2.00	   Simplified the guard:
3.63/2.00	
3.63/2.00	      4: start -> eval : A'=A-B, B'=0, [ B>=1 && A>=B ], cost: 1+B
3.63/2.00	
3.63/2.00	   Solved the limit problem by the following transformations:
3.63/2.00	
3.63/2.00	      Created initial limit problem:
3.63/2.00	
3.63/2.00	      1+B (+), 1+A-B (+/+!), B (+/+!) [not solved]
3.63/2.00	
3.63/2.00	
3.63/2.00	
3.63/2.00	      removing all constraints (solved by SMT)
3.63/2.00	
3.63/2.00	      resulting limit problem: [solved]
3.63/2.00	
3.63/2.00	
3.63/2.00	
3.63/2.00	      applying transformation rule (C) using substitution {A==n,B==n}
3.63/2.00	
3.63/2.00	      resulting limit problem:
3.63/2.00	
3.63/2.00	      [solved]
3.63/2.00	
3.63/2.00	
3.63/2.00	
3.63/2.00	   Solution:
3.63/2.00	
3.63/2.00	      A / n
3.63/2.00	
3.63/2.00	      B / n
3.63/2.00	
3.63/2.00	   Resulting cost 1+n has complexity: Poly(n^1)
3.63/2.00	
3.63/2.00	
3.63/2.00	
3.63/2.00	Found new complexity Poly(n^1).
3.63/2.00	
3.63/2.00	
3.63/2.00	
3.63/2.00	Obtained the following overall complexity (w.r.t. the length of the input n):
3.63/2.00	
3.63/2.00	   Complexity:  Poly(n^1)
3.63/2.00	
3.63/2.00	   Cpx degree:  1
3.63/2.00	
3.63/2.00	   Solved cost: 1+n
3.63/2.00	
3.63/2.00	   Rule cost:   1+B
3.63/2.00	
3.63/2.00	   Rule guard:  [ B>=1 && A>=B ]
3.63/2.00	
3.63/2.00	
3.63/2.00	
3.63/2.00	WORST_CASE(Omega(n^1),?)
3.63/2.00	
3.63/2.00	
3.63/2.00	----------------------------------------
3.63/2.00	
3.63/2.00	(4)
3.63/2.00	BOUNDS(n^1, INF)
3.63/2.02	EOF