3.51/2.20	WORST_CASE(Omega(n^1), O(n^1))
3.51/2.21	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.51/2.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.51/2.21	
3.51/2.21	
3.51/2.21	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + -1 * Arg_1)).
3.51/2.21	
3.51/2.21	(0) CpxIntTrs
3.51/2.21	(1) Koat2 Proof [FINISHED, 130 ms]
3.51/2.21	(2) BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
3.51/2.21	(3) Loat Proof [FINISHED, 215 ms]
3.51/2.21	(4) BOUNDS(n^1, INF)
3.51/2.21	
3.51/2.21	
3.51/2.21	----------------------------------------
3.51/2.21	
3.51/2.21	(0)
3.51/2.21	Obligation:
3.51/2.21	Complexity Int TRS consisting of the following rules:
3.51/2.21	eval(A, B, C) -> Com_1(eval(A, B + 1, C + 1)) :|: A >= B + 1 && A >= C + 1
3.51/2.21	start(A, B, C) -> Com_1(eval(A, B, C)) :|: TRUE
3.51/2.21	
3.51/2.21	The start-symbols are:[start_3]
3.51/2.21	
3.51/2.21	
3.51/2.21	----------------------------------------
3.51/2.21	
3.51/2.21	(1) Koat2 Proof (FINISHED)
3.51/2.21	YES( ?, max([1, 1+Arg_0-Arg_1]) {O(n)})
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	Initial Complexity Problem:
3.51/2.21	
3.51/2.21	  Start:  start  
3.51/2.21	
3.51/2.21	  Program_Vars:  Arg_0, Arg_1, Arg_2  
3.51/2.21	
3.51/2.21	  Temp_Vars:    
3.51/2.21	
3.51/2.21	  Locations:  eval, start  
3.51/2.21	
3.51/2.21	  Transitions:
3.51/2.21	
3.51/2.21	  eval(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1+1,Arg_2+1):|:Arg_1+1 <= Arg_0 && Arg_2+1 <= Arg_0
3.51/2.21	
3.51/2.21	  start(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1,Arg_2):|:  
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	Timebounds: 
3.51/2.21	
3.51/2.21	  Overall timebound: max([1, 1+Arg_0-Arg_1]) {O(n)}
3.51/2.21	
3.51/2.21	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
3.51/2.21	
3.51/2.21	  1: start->eval: 1 {O(1)}
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	Costbounds:
3.51/2.21	
3.51/2.21	  Overall costbound: max([1, 1+Arg_0-Arg_1]) {O(n)}
3.51/2.21	
3.51/2.21	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
3.51/2.21	
3.51/2.21	  1: start->eval: 1 {O(1)}
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	Sizebounds:
3.51/2.21	
3.51/2.21	`Lower:
3.51/2.21	
3.51/2.21	  0: eval->eval, Arg_0: Arg_0 {O(n)}
3.51/2.21	
3.51/2.21	  0: eval->eval, Arg_1: Arg_1 {O(n)}
3.51/2.21	
3.51/2.21	  0: eval->eval, Arg_2: Arg_2 {O(n)}
3.51/2.21	
3.51/2.21	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.51/2.21	
3.51/2.21	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.51/2.21	
3.51/2.21	  1: start->eval, Arg_2: Arg_2 {O(n)}
3.51/2.21	
3.51/2.21	`Upper:
3.51/2.21	
3.51/2.21	  0: eval->eval, Arg_0: Arg_0 {O(n)}
3.51/2.21	
3.51/2.21	  0: eval->eval, Arg_1: Arg_1+max([0, Arg_0-Arg_1]) {O(n)}
3.51/2.21	
3.51/2.21	  0: eval->eval, Arg_2: Arg_2+max([0, Arg_0-Arg_1]) {O(n)}
3.51/2.21	
3.51/2.21	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.51/2.21	
3.51/2.21	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.51/2.21	
3.51/2.21	  1: start->eval, Arg_2: Arg_2 {O(n)}
3.51/2.21	
3.51/2.21	
3.51/2.21	----------------------------------------
3.51/2.21	
3.51/2.21	(2)
3.51/2.21	BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
3.51/2.21	
3.51/2.21	----------------------------------------
3.51/2.21	
3.51/2.21	(3) Loat Proof (FINISHED)
3.51/2.21	
3.51/2.21	
3.51/2.21	### Pre-processing the ITS problem ###
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	Initial linear ITS problem
3.51/2.21	
3.51/2.21	   Start location: start
3.51/2.21	
3.51/2.21	      0: eval -> eval : B'=1+B, C'=1+C, [ A>=1+B && A>=1+C ], cost: 1
3.51/2.21	
3.51/2.21	      1: start -> eval : [], cost: 1
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	### Simplification by acceleration and chaining ###
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	Accelerating simple loops of location 0.
3.51/2.21	
3.51/2.21	   Accelerating the following rules:
3.51/2.21	
3.51/2.21	      0: eval -> eval : B'=1+B, C'=1+C, [ A>=1+B && A>=1+C ], cost: 1
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	   Accelerated rule 0 with metering function A-B (after adding B>=C), yielding the new rule 2.
3.51/2.21	
3.51/2.21	   Accelerated rule 0 with metering function -C+A (after adding B<=C), yielding the new rule 3.
3.51/2.21	
3.51/2.21	   Removing the simple loops: 0.
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	Accelerated all simple loops using metering functions (where possible):
3.51/2.21	
3.51/2.21	   Start location: start
3.51/2.21	
3.51/2.21	      2: eval -> eval : B'=A, C'=C+A-B, [ A>=1+B && A>=1+C && B>=C ], cost: A-B
3.51/2.21	
3.51/2.21	      3: eval -> eval : B'=-C+A+B, C'=A, [ A>=1+B && A>=1+C && B<=C ], cost: -C+A
3.51/2.21	
3.51/2.21	      1: start -> eval : [], cost: 1
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	Chained accelerated rules (with incoming rules):
3.51/2.21	
3.51/2.21	   Start location: start
3.51/2.21	
3.51/2.21	      1: start -> eval : [], cost: 1
3.51/2.21	
3.51/2.21	      4: start -> eval : B'=A, C'=C+A-B, [ A>=1+B && A>=1+C && B>=C ], cost: 1+A-B
3.51/2.21	
3.51/2.21	      5: start -> eval : B'=-C+A+B, C'=A, [ A>=1+B && A>=1+C && B<=C ], cost: 1-C+A
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	Removed unreachable locations (and leaf rules with constant cost):
3.51/2.21	
3.51/2.21	   Start location: start
3.51/2.21	
3.51/2.21	      4: start -> eval : B'=A, C'=C+A-B, [ A>=1+B && A>=1+C && B>=C ], cost: 1+A-B
3.51/2.21	
3.51/2.21	      5: start -> eval : B'=-C+A+B, C'=A, [ A>=1+B && A>=1+C && B<=C ], cost: 1-C+A
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	### Computing asymptotic complexity ###
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	Fully simplified ITS problem
3.51/2.21	
3.51/2.21	   Start location: start
3.51/2.21	
3.51/2.21	      4: start -> eval : B'=A, C'=C+A-B, [ A>=1+B && A>=1+C && B>=C ], cost: 1+A-B
3.51/2.21	
3.51/2.21	      5: start -> eval : B'=-C+A+B, C'=A, [ A>=1+B && A>=1+C && B<=C ], cost: 1-C+A
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	Computing asymptotic complexity for rule 4
3.51/2.21	
3.51/2.21	   Solved the limit problem by the following transformations:
3.51/2.21	
3.51/2.21	      Created initial limit problem:
3.51/2.21	
3.51/2.21	      1-C+B (+/+!), -C+A (+/+!), 1+A-B (+), A-B (+/+!) [not solved]
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	      removing all constraints (solved by SMT)
3.51/2.21	
3.51/2.21	      resulting limit problem: [solved]
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	      applying transformation rule (C) using substitution {C==-n,A==0,B==-n}
3.51/2.21	
3.51/2.21	      resulting limit problem:
3.51/2.21	
3.51/2.21	      [solved]
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	   Solution:
3.51/2.21	
3.51/2.21	      C / -n
3.51/2.21	
3.51/2.21	      A / 0
3.51/2.21	
3.51/2.21	      B / -n
3.51/2.21	
3.51/2.21	   Resulting cost 1+n has complexity: Poly(n^1)
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	Found new complexity Poly(n^1).
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	Obtained the following overall complexity (w.r.t. the length of the input n):
3.51/2.21	
3.51/2.21	   Complexity:  Poly(n^1)
3.51/2.21	
3.51/2.21	   Cpx degree:  1
3.51/2.21	
3.51/2.21	   Solved cost: 1+n
3.51/2.21	
3.51/2.21	   Rule cost:   1+A-B
3.51/2.21	
3.51/2.21	   Rule guard:  [ A>=1+B && A>=1+C && B>=C ]
3.51/2.21	
3.51/2.21	
3.51/2.21	
3.51/2.21	WORST_CASE(Omega(n^1),?)
3.51/2.21	
3.51/2.21	
3.51/2.21	----------------------------------------
3.51/2.21	
3.51/2.21	(4)
3.51/2.21	BOUNDS(n^1, INF)
3.51/2.23	EOF