4.06/2.03	WORST_CASE(Omega(n^1), O(n^1))
4.06/2.04	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.06/2.04	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.06/2.04	
4.06/2.04	
4.06/2.04	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + 3 * Arg_0) + nat(1 + Arg_0) + nat(Arg_1)).
4.06/2.04	
4.06/2.04	(0) CpxIntTrs
4.06/2.04	(1) Koat2 Proof [FINISHED, 187 ms]
4.06/2.04	(2) BOUNDS(1, max(1, 1 + 3 * Arg_0) + nat(1 + Arg_0) + nat(Arg_1))
4.06/2.04	(3) Loat Proof [FINISHED, 353 ms]
4.06/2.04	(4) BOUNDS(n^1, INF)
4.06/2.04	
4.06/2.04	
4.06/2.04	----------------------------------------
4.06/2.04	
4.06/2.04	(0)
4.06/2.04	Obligation:
4.06/2.04	Complexity Int TRS consisting of the following rules:
4.06/2.04	eval1(A, B) -> Com_1(eval2(A, B)) :|: A >= 1
4.06/2.04	eval2(A, B) -> Com_1(eval2(A, B - 1)) :|: A >= 1 && B >= 1
4.06/2.04	eval2(A, B) -> Com_1(eval1(A - 1, B)) :|: A >= 1 && 0 >= B
4.06/2.04	start(A, B) -> Com_1(eval1(A, B)) :|: TRUE
4.06/2.04	
4.06/2.04	The start-symbols are:[start_2]
4.06/2.04	
4.06/2.04	
4.06/2.04	----------------------------------------
4.06/2.04	
4.06/2.04	(1) Koat2 Proof (FINISHED)
4.06/2.04	YES( ?, 1+max([0, 3*Arg_0])+max([0, 1+Arg_0])+max([0, Arg_1]) {O(n)})
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Initial Complexity Problem:
4.06/2.04	
4.06/2.04	  Start:  start  
4.06/2.04	
4.06/2.04	  Program_Vars:  Arg_0, Arg_1  
4.06/2.04	
4.06/2.04	  Temp_Vars:    
4.06/2.04	
4.06/2.04	  Locations:  eval1, eval2, start  
4.06/2.04	
4.06/2.04	  Transitions:
4.06/2.04	
4.06/2.04	  eval1(Arg_0,Arg_1) -> eval2(Arg_0,Arg_1):|:1 <= Arg_0
4.06/2.04	
4.06/2.04	  eval2(Arg_0,Arg_1) -> eval1(Arg_0-1,Arg_1):|:1 <= Arg_0 && 1 <= Arg_0 && Arg_1 <= 0
4.06/2.04	
4.06/2.04	  eval2(Arg_0,Arg_1) -> eval2(Arg_0,Arg_1-1):|:1 <= Arg_0 && 1 <= Arg_0 && 1 <= Arg_1
4.06/2.04	
4.06/2.04	  start(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:  
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Timebounds: 
4.06/2.04	
4.06/2.04	  Overall timebound: 1+max([0, 3*Arg_0])+max([0, 1+Arg_0])+max([0, Arg_1]) {O(n)}
4.06/2.04	
4.06/2.04	  0: eval1->eval2: max([0, 1+Arg_0]) {O(n)}
4.06/2.04	
4.06/2.04	  1: eval2->eval2: max([0, Arg_1]) {O(n)}
4.06/2.04	
4.06/2.04	  2: eval2->eval1: max([0, 3*Arg_0]) {O(n)}
4.06/2.04	
4.06/2.04	  3: start->eval1: 1 {O(1)}
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Costbounds:
4.06/2.04	
4.06/2.04	  Overall costbound: 1+max([0, 3*Arg_0])+max([0, 1+Arg_0])+max([0, Arg_1]) {O(n)}
4.06/2.04	
4.06/2.04	  0: eval1->eval2: max([0, 1+Arg_0]) {O(n)}
4.06/2.04	
4.06/2.04	  1: eval2->eval2: max([0, Arg_1]) {O(n)}
4.06/2.04	
4.06/2.04	  2: eval2->eval1: max([0, 3*Arg_0]) {O(n)}
4.06/2.04	
4.06/2.04	  3: start->eval1: 1 {O(1)}
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Sizebounds:
4.06/2.04	
4.06/2.04	`Lower:
4.06/2.04	
4.06/2.04	  0: eval1->eval2, Arg_0: 1 {O(1)}
4.06/2.04	
4.06/2.04	  0: eval1->eval2, Arg_1: min([0, Arg_1]) {O(n)}
4.06/2.04	
4.06/2.04	  1: eval2->eval2, Arg_0: 1 {O(1)}
4.06/2.04	
4.06/2.04	  1: eval2->eval2, Arg_1: 0 {O(1)}
4.06/2.04	
4.06/2.04	  2: eval2->eval1, Arg_0: 0 {O(1)}
4.06/2.04	
4.06/2.04	  2: eval2->eval1, Arg_1: min([0, Arg_1]) {O(n)}
4.06/2.04	
4.06/2.04	  3: start->eval1, Arg_0: Arg_0 {O(n)}
4.06/2.04	
4.06/2.04	  3: start->eval1, Arg_1: Arg_1 {O(n)}
4.06/2.04	
4.06/2.04	`Upper:
4.06/2.04	
4.06/2.04	  0: eval1->eval2, Arg_0: Arg_0 {O(n)}
4.06/2.04	
4.06/2.04	  0: eval1->eval2, Arg_1: max([0, Arg_1]) {O(n)}
4.06/2.04	
4.06/2.04	  1: eval2->eval2, Arg_0: Arg_0 {O(n)}
4.06/2.04	
4.06/2.04	  1: eval2->eval2, Arg_1: max([0, Arg_1]) {O(n)}
4.06/2.04	
4.06/2.04	  2: eval2->eval1, Arg_0: Arg_0 {O(n)}
4.06/2.04	
4.06/2.04	  2: eval2->eval1, Arg_1: 0 {O(1)}
4.06/2.04	
4.06/2.04	  3: start->eval1, Arg_0: Arg_0 {O(n)}
4.06/2.04	
4.06/2.04	  3: start->eval1, Arg_1: Arg_1 {O(n)}
4.06/2.04	
4.06/2.04	
4.06/2.04	----------------------------------------
4.06/2.04	
4.06/2.04	(2)
4.06/2.04	BOUNDS(1, max(1, 1 + 3 * Arg_0) + nat(1 + Arg_0) + nat(Arg_1))
4.06/2.04	
4.06/2.04	----------------------------------------
4.06/2.04	
4.06/2.04	(3) Loat Proof (FINISHED)
4.06/2.04	
4.06/2.04	
4.06/2.04	### Pre-processing the ITS problem ###
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Initial linear ITS problem
4.06/2.04	
4.06/2.04	   Start location: start
4.06/2.04	
4.06/2.04	      0: eval1 -> eval2 : [ A>=1 ], cost: 1
4.06/2.04	
4.06/2.04	      1: eval2 -> eval2 : B'=-1+B, [ A>=1 && B>=1 ], cost: 1
4.06/2.04	
4.06/2.04	      2: eval2 -> eval1 : A'=-1+A, [ A>=1 && 0>=B ], cost: 1
4.06/2.04	
4.06/2.04	      3: start -> eval1 : [], cost: 1
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	### Simplification by acceleration and chaining ###
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Accelerating simple loops of location 1.
4.06/2.04	
4.06/2.04	   Accelerating the following rules:
4.06/2.04	
4.06/2.04	      1: eval2 -> eval2 : B'=-1+B, [ A>=1 && B>=1 ], cost: 1
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	   Accelerated rule 1 with metering function B, yielding the new rule 4.
4.06/2.04	
4.06/2.04	   Removing the simple loops: 1.
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Accelerated all simple loops using metering functions (where possible):
4.06/2.04	
4.06/2.04	   Start location: start
4.06/2.04	
4.06/2.04	      0: eval1 -> eval2 : [ A>=1 ], cost: 1
4.06/2.04	
4.06/2.04	      2: eval2 -> eval1 : A'=-1+A, [ A>=1 && 0>=B ], cost: 1
4.06/2.04	
4.06/2.04	      4: eval2 -> eval2 : B'=0, [ A>=1 && B>=1 ], cost: B
4.06/2.04	
4.06/2.04	      3: start -> eval1 : [], cost: 1
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Chained accelerated rules (with incoming rules):
4.06/2.04	
4.06/2.04	   Start location: start
4.06/2.04	
4.06/2.04	      0: eval1 -> eval2 : [ A>=1 ], cost: 1
4.06/2.04	
4.06/2.04	      5: eval1 -> eval2 : B'=0, [ A>=1 && B>=1 ], cost: 1+B
4.06/2.04	
4.06/2.04	      2: eval2 -> eval1 : A'=-1+A, [ A>=1 && 0>=B ], cost: 1
4.06/2.04	
4.06/2.04	      3: start -> eval1 : [], cost: 1
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Eliminated locations (on tree-shaped paths):
4.06/2.04	
4.06/2.04	   Start location: start
4.06/2.04	
4.06/2.04	      6: eval1 -> eval1 : A'=-1+A, [ A>=1 && 0>=B ], cost: 2
4.06/2.04	
4.06/2.04	      7: eval1 -> eval1 : A'=-1+A, B'=0, [ A>=1 && B>=1 ], cost: 2+B
4.06/2.04	
4.06/2.04	      3: start -> eval1 : [], cost: 1
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Accelerating simple loops of location 0.
4.06/2.04	
4.06/2.04	   Accelerating the following rules:
4.06/2.04	
4.06/2.04	      6: eval1 -> eval1 : A'=-1+A, [ A>=1 && 0>=B ], cost: 2
4.06/2.04	
4.06/2.04	      7: eval1 -> eval1 : A'=-1+A, B'=0, [ A>=1 && B>=1 ], cost: 2+B
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	   Accelerated rule 6 with metering function A, yielding the new rule 8.
4.06/2.04	
4.06/2.04	   Found no metering function for rule 7.
4.06/2.04	
4.06/2.04	   Removing the simple loops: 6.
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Accelerated all simple loops using metering functions (where possible):
4.06/2.04	
4.06/2.04	   Start location: start
4.06/2.04	
4.06/2.04	      7: eval1 -> eval1 : A'=-1+A, B'=0, [ A>=1 && B>=1 ], cost: 2+B
4.06/2.04	
4.06/2.04	      8: eval1 -> eval1 : A'=0, [ A>=1 && 0>=B ], cost: 2*A
4.06/2.04	
4.06/2.04	      3: start -> eval1 : [], cost: 1
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Chained accelerated rules (with incoming rules):
4.06/2.04	
4.06/2.04	   Start location: start
4.06/2.04	
4.06/2.04	      3: start -> eval1 : [], cost: 1
4.06/2.04	
4.06/2.04	      9: start -> eval1 : A'=-1+A, B'=0, [ A>=1 && B>=1 ], cost: 3+B
4.06/2.04	
4.06/2.04	     10: start -> eval1 : A'=0, [ A>=1 && 0>=B ], cost: 1+2*A
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Removed unreachable locations (and leaf rules with constant cost):
4.06/2.04	
4.06/2.04	   Start location: start
4.06/2.04	
4.06/2.04	      9: start -> eval1 : A'=-1+A, B'=0, [ A>=1 && B>=1 ], cost: 3+B
4.06/2.04	
4.06/2.04	     10: start -> eval1 : A'=0, [ A>=1 && 0>=B ], cost: 1+2*A
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	### Computing asymptotic complexity ###
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Fully simplified ITS problem
4.06/2.04	
4.06/2.04	   Start location: start
4.06/2.04	
4.06/2.04	      9: start -> eval1 : A'=-1+A, B'=0, [ A>=1 && B>=1 ], cost: 3+B
4.06/2.04	
4.06/2.04	     10: start -> eval1 : A'=0, [ A>=1 && 0>=B ], cost: 1+2*A
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Computing asymptotic complexity for rule 9
4.06/2.04	
4.06/2.04	   Solved the limit problem by the following transformations:
4.06/2.04	
4.06/2.04	      Created initial limit problem:
4.06/2.04	
4.06/2.04	      3+B (+), A (+/+!), B (+/+!) [not solved]
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	      removing all constraints (solved by SMT)
4.06/2.04	
4.06/2.04	      resulting limit problem: [solved]
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	      applying transformation rule (C) using substitution {A==n,B==n}
4.06/2.04	
4.06/2.04	      resulting limit problem:
4.06/2.04	
4.06/2.04	      [solved]
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	   Solution:
4.06/2.04	
4.06/2.04	      A / n
4.06/2.04	
4.06/2.04	      B / n
4.06/2.04	
4.06/2.04	   Resulting cost 3+n has complexity: Poly(n^1)
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Found new complexity Poly(n^1).
4.06/2.04	
4.06/2.04	
4.06/2.04	
4.06/2.04	Obtained the following overall complexity (w.r.t. the length of the input n):
4.06/2.05	
4.06/2.05	   Complexity:  Poly(n^1)
4.06/2.05	
4.06/2.05	   Cpx degree:  1
4.06/2.05	
4.06/2.05	   Solved cost: 3+n
4.06/2.05	
4.06/2.05	   Rule cost:   3+B
4.06/2.05	
4.06/2.05	   Rule guard:  [ A>=1 && B>=1 ]
4.06/2.05	
4.06/2.05	
4.06/2.05	
4.06/2.05	WORST_CASE(Omega(n^1),?)
4.06/2.05	
4.06/2.05	
4.06/2.05	----------------------------------------
4.06/2.05	
4.06/2.05	(4)
4.06/2.05	BOUNDS(n^1, INF)
4.06/2.07	EOF