4.24/2.31	WORST_CASE(Omega(n^1), O(n^1))
4.24/2.32	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.24/2.32	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.24/2.32	
4.24/2.32	
4.24/2.32	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + -1 * Arg_0 + Arg_1) + nat(Arg_0 + -1 * Arg_1)).
4.24/2.32	
4.24/2.32	(0) CpxIntTrs
4.24/2.32	(1) Koat2 Proof [FINISHED, 143 ms]
4.24/2.32	(2) BOUNDS(1, max(1, 1 + -1 * Arg_0 + Arg_1) + nat(Arg_0 + -1 * Arg_1))
4.24/2.32	(3) Loat Proof [FINISHED, 326 ms]
4.24/2.32	(4) BOUNDS(n^1, INF)
4.24/2.32	
4.24/2.32	
4.24/2.32	----------------------------------------
4.24/2.32	
4.24/2.32	(0)
4.24/2.32	Obligation:
4.24/2.32	Complexity Int TRS consisting of the following rules:
4.24/2.32	eval(A, B) -> Com_1(eval(A, B + 1)) :|: A >= B + 1
4.24/2.32	eval(A, B) -> Com_1(eval(A, B + 1)) :|: B >= A + 1 && A >= B + 1
4.24/2.32	eval(A, B) -> Com_1(eval(A + 1, B)) :|: A >= B + 1 && B >= A
4.24/2.32	eval(A, B) -> Com_1(eval(A + 1, B)) :|: B >= A + 1 && B >= A
4.24/2.32	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
4.24/2.32	
4.24/2.32	The start-symbols are:[start_2]
4.24/2.32	
4.24/2.32	
4.24/2.32	----------------------------------------
4.24/2.32	
4.24/2.32	(1) Koat2 Proof (FINISHED)
4.24/2.32	YES( ?, 1+max([0, Arg_1-Arg_0])+max([0, Arg_0-Arg_1]) {O(n)})
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Initial Complexity Problem:
4.24/2.32	
4.24/2.32	  Start:  start  
4.24/2.32	
4.24/2.32	  Program_Vars:  Arg_0, Arg_1  
4.24/2.32	
4.24/2.32	  Temp_Vars:    
4.24/2.32	
4.24/2.32	  Locations:  eval, start  
4.24/2.32	
4.24/2.32	  Transitions:
4.24/2.32	
4.24/2.32	  eval(Arg_0,Arg_1) -> eval(Arg_0,Arg_1+1):|:Arg_1+1 <= Arg_0
4.24/2.32	
4.24/2.32	  eval(Arg_0,Arg_1) -> eval(Arg_0+1,Arg_1):|:Arg_0+1 <= Arg_1 && Arg_0 <= Arg_1
4.24/2.32	
4.24/2.32	  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Timebounds: 
4.24/2.32	
4.24/2.32	  Overall timebound: 1+max([0, Arg_1-Arg_0])+max([0, Arg_0-Arg_1]) {O(n)}
4.24/2.32	
4.24/2.32	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
4.24/2.32	
4.24/2.32	  3: eval->eval: max([0, Arg_1-Arg_0]) {O(n)}
4.24/2.32	
4.24/2.32	  4: start->eval: 1 {O(1)}
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Costbounds:
4.24/2.32	
4.24/2.32	  Overall costbound: 1+max([0, Arg_1-Arg_0])+max([0, Arg_0-Arg_1]) {O(n)}
4.24/2.32	
4.24/2.32	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
4.24/2.32	
4.24/2.32	  3: eval->eval: max([0, Arg_1-Arg_0]) {O(n)}
4.24/2.32	
4.24/2.32	  4: start->eval: 1 {O(1)}
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Sizebounds:
4.24/2.32	
4.24/2.32	`Lower:
4.24/2.32	
4.24/2.32	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.24/2.32	
4.24/2.32	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.24/2.32	
4.24/2.32	  3: eval->eval, Arg_0: Arg_0 {O(n)}
4.24/2.32	
4.24/2.32	  3: eval->eval, Arg_1: Arg_1 {O(n)}
4.24/2.32	
4.24/2.32	  4: start->eval, Arg_0: Arg_0 {O(n)}
4.24/2.32	
4.24/2.32	  4: start->eval, Arg_1: Arg_1 {O(n)}
4.24/2.32	
4.24/2.32	`Upper:
4.24/2.32	
4.24/2.32	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.24/2.32	
4.24/2.32	  0: eval->eval, Arg_1: Arg_1+max([0, Arg_0-Arg_1]) {O(n)}
4.24/2.32	
4.24/2.32	  3: eval->eval, Arg_0: Arg_0+max([0, Arg_1-Arg_0]) {O(n)}
4.24/2.32	
4.24/2.32	  3: eval->eval, Arg_1: Arg_1 {O(n)}
4.24/2.32	
4.24/2.32	  4: start->eval, Arg_0: Arg_0 {O(n)}
4.24/2.32	
4.24/2.32	  4: start->eval, Arg_1: Arg_1 {O(n)}
4.24/2.32	
4.24/2.32	
4.24/2.32	----------------------------------------
4.24/2.32	
4.24/2.32	(2)
4.24/2.32	BOUNDS(1, max(1, 1 + -1 * Arg_0 + Arg_1) + nat(Arg_0 + -1 * Arg_1))
4.24/2.32	
4.24/2.32	----------------------------------------
4.24/2.32	
4.24/2.32	(3) Loat Proof (FINISHED)
4.24/2.32	
4.24/2.32	
4.24/2.32	### Pre-processing the ITS problem ###
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Initial linear ITS problem
4.24/2.32	
4.24/2.32	   Start location: start
4.24/2.32	
4.24/2.32	      0: eval -> eval : B'=1+B, [ A>=1+B ], cost: 1
4.24/2.32	
4.24/2.32	      1: eval -> eval : B'=1+B, [ B>=1+A && A>=1+B ], cost: 1
4.24/2.32	
4.24/2.32	      2: eval -> eval : A'=1+A, [ A>=1+B && B>=A ], cost: 1
4.24/2.32	
4.24/2.32	      3: eval -> eval : A'=1+A, [ B>=1+A && B>=A ], cost: 1
4.24/2.32	
4.24/2.32	      4: start -> eval : [], cost: 1
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Removed rules with unsatisfiable guard:
4.24/2.32	
4.24/2.32	   Start location: start
4.24/2.32	
4.24/2.32	      0: eval -> eval : B'=1+B, [ A>=1+B ], cost: 1
4.24/2.32	
4.24/2.32	      3: eval -> eval : A'=1+A, [ B>=1+A && B>=A ], cost: 1
4.24/2.32	
4.24/2.32	      4: start -> eval : [], cost: 1
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Simplified all rules, resulting in:
4.24/2.32	
4.24/2.32	   Start location: start
4.24/2.32	
4.24/2.32	      0: eval -> eval : B'=1+B, [ A>=1+B ], cost: 1
4.24/2.32	
4.24/2.32	      3: eval -> eval : A'=1+A, [ B>=1+A ], cost: 1
4.24/2.32	
4.24/2.32	      4: start -> eval : [], cost: 1
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	### Simplification by acceleration and chaining ###
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Accelerating simple loops of location 0.
4.24/2.32	
4.24/2.32	   Accelerating the following rules:
4.24/2.32	
4.24/2.32	      0: eval -> eval : B'=1+B, [ A>=1+B ], cost: 1
4.24/2.32	
4.24/2.32	      3: eval -> eval : A'=1+A, [ B>=1+A ], cost: 1
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	   Accelerated rule 0 with metering function A-B, yielding the new rule 5.
4.24/2.32	
4.24/2.32	   Accelerated rule 3 with metering function -A+B, yielding the new rule 6.
4.24/2.32	
4.24/2.32	   Removing the simple loops: 0 3.
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Accelerated all simple loops using metering functions (where possible):
4.24/2.32	
4.24/2.32	   Start location: start
4.24/2.32	
4.24/2.32	      5: eval -> eval : B'=A, [ A>=1+B ], cost: A-B
4.24/2.32	
4.24/2.32	      6: eval -> eval : A'=B, [ B>=1+A ], cost: -A+B
4.24/2.32	
4.24/2.32	      4: start -> eval : [], cost: 1
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Chained accelerated rules (with incoming rules):
4.24/2.32	
4.24/2.32	   Start location: start
4.24/2.32	
4.24/2.32	      4: start -> eval : [], cost: 1
4.24/2.32	
4.24/2.32	      7: start -> eval : B'=A, [ A>=1+B ], cost: 1+A-B
4.24/2.32	
4.24/2.32	      8: start -> eval : A'=B, [ B>=1+A ], cost: 1-A+B
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Removed unreachable locations (and leaf rules with constant cost):
4.24/2.32	
4.24/2.32	   Start location: start
4.24/2.32	
4.24/2.32	      7: start -> eval : B'=A, [ A>=1+B ], cost: 1+A-B
4.24/2.32	
4.24/2.32	      8: start -> eval : A'=B, [ B>=1+A ], cost: 1-A+B
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	### Computing asymptotic complexity ###
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Fully simplified ITS problem
4.24/2.32	
4.24/2.32	   Start location: start
4.24/2.32	
4.24/2.32	      7: start -> eval : B'=A, [ A>=1+B ], cost: 1+A-B
4.24/2.32	
4.24/2.32	      8: start -> eval : A'=B, [ B>=1+A ], cost: 1-A+B
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Computing asymptotic complexity for rule 7
4.24/2.32	
4.24/2.32	   Solved the limit problem by the following transformations:
4.24/2.32	
4.24/2.32	      Created initial limit problem:
4.24/2.32	
4.24/2.32	      1+A-B (+), A-B (+/+!) [not solved]
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	      removing all constraints (solved by SMT)
4.24/2.32	
4.24/2.32	      resulting limit problem: [solved]
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	      applying transformation rule (C) using substitution {A==0,B==-n}
4.24/2.32	
4.24/2.32	      resulting limit problem:
4.24/2.32	
4.24/2.32	      [solved]
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	   Solution:
4.24/2.32	
4.24/2.32	      A / 0
4.24/2.32	
4.24/2.32	      B / -n
4.24/2.32	
4.24/2.32	   Resulting cost 1+n has complexity: Poly(n^1)
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Found new complexity Poly(n^1).
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	Obtained the following overall complexity (w.r.t. the length of the input n):
4.24/2.32	
4.24/2.32	   Complexity:  Poly(n^1)
4.24/2.32	
4.24/2.32	   Cpx degree:  1
4.24/2.32	
4.24/2.32	   Solved cost: 1+n
4.24/2.32	
4.24/2.32	   Rule cost:   1+A-B
4.24/2.32	
4.24/2.32	   Rule guard:  [ A>=1+B ]
4.24/2.32	
4.24/2.32	
4.24/2.32	
4.24/2.32	WORST_CASE(Omega(n^1),?)
4.24/2.32	
4.24/2.32	
4.24/2.32	----------------------------------------
4.24/2.32	
4.24/2.32	(4)
4.24/2.32	BOUNDS(n^1, INF)
4.24/2.34	EOF