4.44/2.33	WORST_CASE(Omega(n^1), O(n^1))
4.44/2.34	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.44/2.34	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.44/2.34	
4.44/2.34	
4.44/2.34	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_1) + nat(Arg_0)).
4.44/2.34	
4.44/2.34	(0) CpxIntTrs
4.44/2.34	(1) Koat2 Proof [FINISHED, 201 ms]
4.44/2.34	(2) BOUNDS(1, max(1, 1 + Arg_1) + nat(Arg_0))
4.44/2.34	(3) Loat Proof [FINISHED, 603 ms]
4.44/2.34	(4) BOUNDS(n^1, INF)
4.44/2.34	
4.44/2.34	
4.44/2.34	----------------------------------------
4.44/2.34	
4.44/2.34	(0)
4.44/2.34	Obligation:
4.44/2.34	Complexity Int TRS consisting of the following rules:
4.44/2.34	eval(A, B) -> Com_1(eval(A - B, B)) :|: A >= B + 1 && A >= 1 && B >= 1
4.44/2.34	eval(A, B) -> Com_1(eval(A - B, B)) :|: B >= A + 1 && A >= 1 && B >= 1 && A >= B + 1
4.44/2.34	eval(A, B) -> Com_1(eval(A, B - A)) :|: A >= B + 1 && A >= 1 && B >= 1 && B >= A
4.44/2.34	eval(A, B) -> Com_1(eval(A, B - A)) :|: B >= A + 1 && A >= 1 && B >= 1 && B >= A
4.44/2.34	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
4.44/2.34	
4.44/2.34	The start-symbols are:[start_2]
4.44/2.34	
4.44/2.34	
4.44/2.34	----------------------------------------
4.44/2.34	
4.44/2.34	(1) Koat2 Proof (FINISHED)
4.44/2.34	YES( ?, 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)})
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Initial Complexity Problem:
4.44/2.34	
4.44/2.34	  Start:  start  
4.44/2.34	
4.44/2.34	  Program_Vars:  Arg_0, Arg_1  
4.44/2.34	
4.44/2.34	  Temp_Vars:    
4.44/2.34	
4.44/2.34	  Locations:  eval, start  
4.44/2.34	
4.44/2.34	  Transitions:
4.44/2.34	
4.44/2.34	  eval(Arg_0,Arg_1) -> eval(Arg_0-Arg_1,Arg_1):|:Arg_1+1 <= Arg_0 && 1 <= Arg_0 && 1 <= Arg_1
4.44/2.34	
4.44/2.34	  eval(Arg_0,Arg_1) -> eval(Arg_0,Arg_1-Arg_0):|:Arg_0+1 <= Arg_1 && 1 <= Arg_0 && 1 <= Arg_1 && Arg_0 <= Arg_1
4.44/2.34	
4.44/2.34	  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Timebounds: 
4.44/2.34	
4.44/2.34	  Overall timebound: 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)}
4.44/2.34	
4.44/2.34	  0: eval->eval: max([0, Arg_0]) {O(n)}
4.44/2.34	
4.44/2.34	  3: eval->eval: max([0, Arg_1]) {O(n)}
4.44/2.34	
4.44/2.34	  4: start->eval: 1 {O(1)}
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Costbounds:
4.44/2.34	
4.44/2.34	  Overall costbound: 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)}
4.44/2.34	
4.44/2.34	  0: eval->eval: max([0, Arg_0]) {O(n)}
4.44/2.34	
4.44/2.34	  3: eval->eval: max([0, Arg_1]) {O(n)}
4.44/2.34	
4.44/2.34	  4: start->eval: 1 {O(1)}
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Sizebounds:
4.44/2.34	
4.44/2.34	`Lower:
4.44/2.34	
4.44/2.34	  0: eval->eval, Arg_0: 1 {O(1)}
4.44/2.34	
4.44/2.34	  0: eval->eval, Arg_1: 1 {O(1)}
4.44/2.34	
4.44/2.34	  3: eval->eval, Arg_0: 1 {O(1)}
4.44/2.34	
4.44/2.34	  3: eval->eval, Arg_1: 1 {O(1)}
4.44/2.34	
4.44/2.34	  4: start->eval, Arg_0: Arg_0 {O(n)}
4.44/2.34	
4.44/2.34	  4: start->eval, Arg_1: Arg_1 {O(n)}
4.44/2.34	
4.44/2.34	`Upper:
4.44/2.34	
4.44/2.34	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.44/2.34	
4.44/2.34	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.44/2.34	
4.44/2.34	  3: eval->eval, Arg_0: Arg_0 {O(n)}
4.44/2.34	
4.44/2.34	  3: eval->eval, Arg_1: Arg_1 {O(n)}
4.44/2.34	
4.44/2.34	  4: start->eval, Arg_0: Arg_0 {O(n)}
4.44/2.34	
4.44/2.34	  4: start->eval, Arg_1: Arg_1 {O(n)}
4.44/2.34	
4.44/2.34	
4.44/2.34	----------------------------------------
4.44/2.34	
4.44/2.34	(2)
4.44/2.34	BOUNDS(1, max(1, 1 + Arg_1) + nat(Arg_0))
4.44/2.34	
4.44/2.34	----------------------------------------
4.44/2.34	
4.44/2.34	(3) Loat Proof (FINISHED)
4.44/2.34	
4.44/2.34	
4.44/2.34	### Pre-processing the ITS problem ###
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Initial linear ITS problem
4.44/2.34	
4.44/2.34	   Start location: start
4.44/2.34	
4.44/2.34	      0: eval -> eval : A'=A-B, [ A>=1+B && A>=1 && B>=1 ], cost: 1
4.44/2.34	
4.44/2.34	      1: eval -> eval : A'=A-B, [ B>=1+A && A>=1 && B>=1 && A>=1+B ], cost: 1
4.44/2.34	
4.44/2.34	      2: eval -> eval : B'=-A+B, [ A>=1+B && A>=1 && B>=1 && B>=A ], cost: 1
4.44/2.34	
4.44/2.34	      3: eval -> eval : B'=-A+B, [ B>=1+A && A>=1 && B>=1 && B>=A ], cost: 1
4.44/2.34	
4.44/2.34	      4: start -> eval : [], cost: 1
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Removed rules with unsatisfiable guard:
4.44/2.34	
4.44/2.34	   Start location: start
4.44/2.34	
4.44/2.34	      0: eval -> eval : A'=A-B, [ A>=1+B && A>=1 && B>=1 ], cost: 1
4.44/2.34	
4.44/2.34	      3: eval -> eval : B'=-A+B, [ B>=1+A && A>=1 && B>=1 && B>=A ], cost: 1
4.44/2.34	
4.44/2.34	      4: start -> eval : [], cost: 1
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Simplified all rules, resulting in:
4.44/2.34	
4.44/2.34	   Start location: start
4.44/2.34	
4.44/2.34	      0: eval -> eval : A'=A-B, [ A>=1+B && A>=1 && B>=1 ], cost: 1
4.44/2.34	
4.44/2.34	      3: eval -> eval : B'=-A+B, [ B>=1+A && A>=1 ], cost: 1
4.44/2.34	
4.44/2.34	      4: start -> eval : [], cost: 1
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	### Simplification by acceleration and chaining ###
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Accelerating simple loops of location 0.
4.44/2.34	
4.44/2.34	   Accelerating the following rules:
4.44/2.34	
4.44/2.34	      0: eval -> eval : A'=A-B, [ A>=1+B && A>=1 && B>=1 ], cost: 1
4.44/2.34	
4.44/2.34	      3: eval -> eval : B'=-A+B, [ B>=1+A && A>=1 ], cost: 1
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	   Accelerated rule 0 with backward acceleration, yielding the new rule 5.
4.44/2.34	
4.44/2.34	   Accelerated rule 3 with backward acceleration, yielding the new rule 6.
4.44/2.34	
4.44/2.34	   Removing the simple loops: 0 3.
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Accelerated all simple loops using metering functions (where possible):
4.44/2.34	
4.44/2.34	   Start location: start
4.44/2.34	
4.44/2.34	      5: eval -> eval : A'=-k*B+A, [ A>=1+B && A>=1 && B>=1 && k>0 && -(-1+k)*B+A>=1+B && -(-1+k)*B+A>=1 ], cost: k
4.44/2.34	
4.44/2.34	      6: eval -> eval : B'=B-A*k_1, [ B>=1+A && A>=1 && k_1>0 && -A*(-1+k_1)+B>=1+A ], cost: k_1
4.44/2.34	
4.44/2.34	      4: start -> eval : [], cost: 1
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Chained accelerated rules (with incoming rules):
4.44/2.34	
4.44/2.34	   Start location: start
4.44/2.34	
4.44/2.34	      4: start -> eval : [], cost: 1
4.44/2.34	
4.44/2.34	      7: start -> eval : A'=-k*B+A, [ A>=1+B && A>=1 && B>=1 && k>0 && -(-1+k)*B+A>=1+B && -(-1+k)*B+A>=1 ], cost: 1+k
4.44/2.34	
4.44/2.34	      8: start -> eval : B'=B-A*k_1, [ B>=1+A && A>=1 && k_1>0 && -A*(-1+k_1)+B>=1+A ], cost: 1+k_1
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Removed unreachable locations (and leaf rules with constant cost):
4.44/2.34	
4.44/2.34	   Start location: start
4.44/2.34	
4.44/2.34	      7: start -> eval : A'=-k*B+A, [ A>=1+B && A>=1 && B>=1 && k>0 && -(-1+k)*B+A>=1+B && -(-1+k)*B+A>=1 ], cost: 1+k
4.44/2.34	
4.44/2.34	      8: start -> eval : B'=B-A*k_1, [ B>=1+A && A>=1 && k_1>0 && -A*(-1+k_1)+B>=1+A ], cost: 1+k_1
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	### Computing asymptotic complexity ###
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Fully simplified ITS problem
4.44/2.34	
4.44/2.34	   Start location: start
4.44/2.34	
4.44/2.34	      7: start -> eval : A'=-k*B+A, [ A>=1+B && A>=1 && B>=1 && k>0 && -(-1+k)*B+A>=1+B && -(-1+k)*B+A>=1 ], cost: 1+k
4.44/2.34	
4.44/2.34	      8: start -> eval : B'=B-A*k_1, [ B>=1+A && A>=1 && k_1>0 && -A*(-1+k_1)+B>=1+A ], cost: 1+k_1
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Computing asymptotic complexity for rule 7
4.44/2.34	
4.44/2.34	   Simplified the guard:
4.44/2.34	
4.44/2.34	      7: start -> eval : A'=-k*B+A, [ B>=1 && k>0 && -(-1+k)*B+A>=1+B ], cost: 1+k
4.44/2.34	
4.44/2.34	   Solved the limit problem by the following transformations:
4.44/2.34	
4.44/2.34	      Created initial limit problem:
4.44/2.34	
4.44/2.34	      -(-1+k)*B+A-B (+/+!), k (+/+!), 1+k (+), B (+/+!) [not solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	      removing all constraints (solved by SMT)
4.44/2.34	
4.44/2.34	      resulting limit problem: [solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	      applying transformation rule (C) using substitution {k==n,A==2*n,B==1}
4.44/2.34	
4.44/2.34	      resulting limit problem:
4.44/2.34	
4.44/2.34	      [solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	   Solved the limit problem by the following transformations:
4.44/2.34	
4.44/2.34	      Created initial limit problem:
4.44/2.34	
4.44/2.34	      -(-1+k)*B+A-B (+/+!), k (+/+!), 1+k (+), B (+/+!) [not solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	      applying transformation rule (C) using substitution {B==1}
4.44/2.34	
4.44/2.34	      resulting limit problem:
4.44/2.34	
4.44/2.34	      1 (+/+!), k (+/+!), 1+k (+), -k+A (+/+!) [not solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	      applying transformation rule (B), deleting 1 (+/+!)
4.44/2.34	
4.44/2.34	      resulting limit problem:
4.44/2.34	
4.44/2.34	      k (+/+!), 1+k (+), -k+A (+/+!) [not solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	      removing all constraints (solved by SMT)
4.44/2.34	
4.44/2.34	      resulting limit problem: [solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	      applying transformation rule (C) using substitution {k==n,A==2*n}
4.44/2.34	
4.44/2.34	      resulting limit problem:
4.44/2.34	
4.44/2.34	      [solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	   Solution:
4.44/2.34	
4.44/2.34	      k / n
4.44/2.34	
4.44/2.34	      A / 2*n
4.44/2.34	
4.44/2.34	      B / 1
4.44/2.34	
4.44/2.34	   Resulting cost 1+n has complexity: Poly(n^1)
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Found new complexity Poly(n^1).
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Computing asymptotic complexity for rule 8
4.44/2.34	
4.44/2.34	   Simplified the guard:
4.44/2.34	
4.44/2.34	      8: start -> eval : B'=B-A*k_1, [ A>=1 && k_1>0 && -A*(-1+k_1)+B>=1+A ], cost: 1+k_1
4.44/2.34	
4.44/2.34	   Solved the limit problem by the following transformations:
4.44/2.34	
4.44/2.34	      Created initial limit problem:
4.44/2.34	
4.44/2.34	      1+k_1 (+), -A*(-1+k_1)-A+B (+/+!), A (+/+!), k_1 (+/+!) [not solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	      removing all constraints (solved by SMT)
4.44/2.34	
4.44/2.34	      resulting limit problem: [solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	      applying transformation rule (C) using substitution {A==1,k_1==n,B==2*n}
4.44/2.34	
4.44/2.34	      resulting limit problem:
4.44/2.34	
4.44/2.34	      [solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	   Solved the limit problem by the following transformations:
4.44/2.34	
4.44/2.34	      Created initial limit problem:
4.44/2.34	
4.44/2.34	      1+k_1 (+), -A*(-1+k_1)-A+B (+/+!), A (+/+!), k_1 (+/+!) [not solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	      applying transformation rule (C) using substitution {A==1}
4.44/2.34	
4.44/2.34	      resulting limit problem:
4.44/2.34	
4.44/2.34	      1 (+/+!), 1+k_1 (+), -k_1+B (+/+!), k_1 (+/+!) [not solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	      applying transformation rule (B), deleting 1 (+/+!)
4.44/2.34	
4.44/2.34	      resulting limit problem:
4.44/2.34	
4.44/2.34	      1+k_1 (+), -k_1+B (+/+!), k_1 (+/+!) [not solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	      removing all constraints (solved by SMT)
4.44/2.34	
4.44/2.34	      resulting limit problem: [solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	      applying transformation rule (C) using substitution {k_1==n,B==2*n}
4.44/2.34	
4.44/2.34	      resulting limit problem:
4.44/2.34	
4.44/2.34	      [solved]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	   Solution:
4.44/2.34	
4.44/2.34	      A / 1
4.44/2.34	
4.44/2.34	      k_1 / n
4.44/2.34	
4.44/2.34	      B / 2*n
4.44/2.34	
4.44/2.34	   Resulting cost 1+n has complexity: Poly(n^1)
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	Obtained the following overall complexity (w.r.t. the length of the input n):
4.44/2.34	
4.44/2.34	   Complexity:  Poly(n^1)
4.44/2.34	
4.44/2.34	   Cpx degree:  1
4.44/2.34	
4.44/2.34	   Solved cost: 1+n
4.44/2.34	
4.44/2.34	   Rule cost:   1+k
4.44/2.34	
4.44/2.34	   Rule guard:  [ B>=1 && k>0 && -(-1+k)*B+A>=1+B ]
4.44/2.34	
4.44/2.34	
4.44/2.34	
4.44/2.34	WORST_CASE(Omega(n^1),?)
4.44/2.34	
4.44/2.34	
4.44/2.34	----------------------------------------
4.44/2.34	
4.44/2.34	(4)
4.44/2.34	BOUNDS(n^1, INF)
4.44/2.37	EOF