4.04/1.96	WORST_CASE(Omega(n^1), O(n^1))
4.04/1.97	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.04/1.97	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.04/1.97	
4.04/1.97	
4.04/1.97	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + 2 * Arg_0) + nat(Arg_0)).
4.04/1.97	
4.04/1.97	(0) CpxIntTrs
4.04/1.97	(1) Koat2 Proof [FINISHED, 216 ms]
4.04/1.97	(2) BOUNDS(1, max(1, 1 + 2 * Arg_0) + nat(Arg_0))
4.04/1.97	(3) Loat Proof [FINISHED, 307 ms]
4.04/1.97	(4) BOUNDS(n^1, INF)
4.04/1.97	
4.04/1.97	
4.04/1.97	----------------------------------------
4.04/1.97	
4.04/1.97	(0)
4.04/1.97	Obligation:
4.04/1.97	Complexity Int TRS consisting of the following rules:
4.04/1.97	eval1(A, B) -> Com_1(eval2(A, B)) :|: A >= 1 && B >= A && B <= A
4.04/1.97	eval2(A, B) -> Com_1(eval2(A - 1, B - 1)) :|: A >= 1
4.04/1.97	eval2(A, B) -> Com_1(eval1(A, B)) :|: 0 >= A
4.04/1.97	start(A, B) -> Com_1(eval1(A, B)) :|: TRUE
4.04/1.97	
4.04/1.97	The start-symbols are:[start_2]
4.04/1.97	
4.04/1.97	
4.04/1.97	----------------------------------------
4.04/1.97	
4.04/1.97	(1) Koat2 Proof (FINISHED)
4.04/1.97	YES( ?, 1+2*max([0, Arg_0])+max([0, Arg_0]) {O(n)})
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Initial Complexity Problem:
4.04/1.97	
4.04/1.97	  Start:  start  
4.04/1.97	
4.04/1.97	  Program_Vars:  Arg_0, Arg_1  
4.04/1.97	
4.04/1.97	  Temp_Vars:    
4.04/1.97	
4.04/1.97	  Locations:  eval1, eval2, start  
4.04/1.97	
4.04/1.97	  Transitions:
4.04/1.97	
4.04/1.97	  eval1(Arg_0,Arg_1) -> eval2(Arg_0,Arg_1):|:1 <= Arg_0 && Arg_1 <= Arg_0 && Arg_0 <= Arg_1
4.04/1.97	
4.04/1.97	  eval2(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:Arg_1 <= Arg_0 && Arg_0 <= Arg_1 && Arg_0 <= 0
4.04/1.97	
4.04/1.97	  eval2(Arg_0,Arg_1) -> eval2(Arg_0-1,Arg_1-1):|:Arg_1 <= Arg_0 && Arg_0 <= Arg_1 && 1 <= Arg_0
4.04/1.97	
4.04/1.97	  start(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:  
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Timebounds: 
4.04/1.97	
4.04/1.97	  Overall timebound: 1+2*max([0, Arg_0])+max([0, Arg_0]) {O(n)}
4.04/1.97	
4.04/1.97	  0: eval1->eval2: max([0, Arg_0]) {O(n)}
4.04/1.97	
4.04/1.97	  1: eval2->eval2: max([0, Arg_0]) {O(n)}
4.04/1.97	
4.04/1.97	  2: eval2->eval1: max([0, Arg_0]) {O(n)}
4.04/1.97	
4.04/1.97	  3: start->eval1: 1 {O(1)}
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Costbounds:
4.04/1.97	
4.04/1.97	  Overall costbound: 1+2*max([0, Arg_0])+max([0, Arg_0]) {O(n)}
4.04/1.97	
4.04/1.97	  0: eval1->eval2: max([0, Arg_0]) {O(n)}
4.04/1.97	
4.04/1.97	  1: eval2->eval2: max([0, Arg_0]) {O(n)}
4.04/1.97	
4.04/1.97	  2: eval2->eval1: max([0, Arg_0]) {O(n)}
4.04/1.97	
4.04/1.97	  3: start->eval1: 1 {O(1)}
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Sizebounds:
4.04/1.97	
4.04/1.97	`Lower:
4.04/1.97	
4.04/1.97	  0: eval1->eval2, Arg_0: 1 {O(1)}
4.04/1.97	
4.04/1.97	  0: eval1->eval2, Arg_1: 1 {O(1)}
4.04/1.97	
4.04/1.97	  1: eval2->eval2, Arg_0: 0 {O(1)}
4.04/1.97	
4.04/1.97	  1: eval2->eval2, Arg_1: 0 {O(1)}
4.04/1.97	
4.04/1.97	  2: eval2->eval1, Arg_0: 0 {O(1)}
4.04/1.97	
4.04/1.97	  2: eval2->eval1, Arg_1: 0 {O(1)}
4.04/1.97	
4.04/1.97	  3: start->eval1, Arg_0: Arg_0 {O(n)}
4.04/1.97	
4.04/1.97	  3: start->eval1, Arg_1: Arg_1 {O(n)}
4.04/1.97	
4.04/1.97	`Upper:
4.04/1.97	
4.04/1.97	  0: eval1->eval2, Arg_0: Arg_0 {O(n)}
4.04/1.97	
4.04/1.97	  0: eval1->eval2, Arg_1: Arg_1 {O(n)}
4.04/1.97	
4.04/1.97	  1: eval2->eval2, Arg_0: Arg_0 {O(n)}
4.04/1.97	
4.04/1.97	  1: eval2->eval2, Arg_1: Arg_1 {O(n)}
4.04/1.97	
4.04/1.97	  2: eval2->eval1, Arg_0: 0 {O(1)}
4.04/1.97	
4.04/1.97	  2: eval2->eval1, Arg_1: 0 {O(1)}
4.04/1.97	
4.04/1.97	  3: start->eval1, Arg_0: Arg_0 {O(n)}
4.04/1.97	
4.04/1.97	  3: start->eval1, Arg_1: Arg_1 {O(n)}
4.04/1.97	
4.04/1.97	
4.04/1.97	----------------------------------------
4.04/1.97	
4.04/1.97	(2)
4.04/1.97	BOUNDS(1, max(1, 1 + 2 * Arg_0) + nat(Arg_0))
4.04/1.97	
4.04/1.97	----------------------------------------
4.04/1.97	
4.04/1.97	(3) Loat Proof (FINISHED)
4.04/1.97	
4.04/1.97	
4.04/1.97	### Pre-processing the ITS problem ###
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Initial linear ITS problem
4.04/1.97	
4.04/1.97	   Start location: start
4.04/1.97	
4.04/1.97	      0: eval1 -> eval2 : [ A>=1 && B==A ], cost: 1
4.04/1.97	
4.04/1.97	      1: eval2 -> eval2 : A'=-1+A, B'=-1+B, [ A>=1 ], cost: 1
4.04/1.97	
4.04/1.97	      2: eval2 -> eval1 : [ 0>=A ], cost: 1
4.04/1.97	
4.04/1.97	      3: start -> eval1 : [], cost: 1
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	### Simplification by acceleration and chaining ###
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Accelerating simple loops of location 1.
4.04/1.97	
4.04/1.97	   Accelerating the following rules:
4.04/1.97	
4.04/1.97	      1: eval2 -> eval2 : A'=-1+A, B'=-1+B, [ A>=1 ], cost: 1
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	   Accelerated rule 1 with metering function A, yielding the new rule 4.
4.04/1.97	
4.04/1.97	   Removing the simple loops: 1.
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Accelerated all simple loops using metering functions (where possible):
4.04/1.97	
4.04/1.97	   Start location: start
4.04/1.97	
4.04/1.97	      0: eval1 -> eval2 : [ A>=1 && B==A ], cost: 1
4.04/1.97	
4.04/1.97	      2: eval2 -> eval1 : [ 0>=A ], cost: 1
4.04/1.97	
4.04/1.97	      4: eval2 -> eval2 : A'=0, B'=-A+B, [ A>=1 ], cost: A
4.04/1.97	
4.04/1.97	      3: start -> eval1 : [], cost: 1
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Chained accelerated rules (with incoming rules):
4.04/1.97	
4.04/1.97	   Start location: start
4.04/1.97	
4.04/1.97	      0: eval1 -> eval2 : [ A>=1 && B==A ], cost: 1
4.04/1.97	
4.04/1.97	      5: eval1 -> eval2 : A'=0, B'=-A+B, [ A>=1 && B==A ], cost: 1+A
4.04/1.97	
4.04/1.97	      2: eval2 -> eval1 : [ 0>=A ], cost: 1
4.04/1.97	
4.04/1.97	      3: start -> eval1 : [], cost: 1
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Eliminated locations (on tree-shaped paths):
4.04/1.97	
4.04/1.97	   Start location: start
4.04/1.97	
4.04/1.97	      6: eval1 -> eval1 : A'=0, B'=-A+B, [ A>=1 && B==A ], cost: 2+A
4.04/1.97	
4.04/1.97	      3: start -> eval1 : [], cost: 1
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Accelerating simple loops of location 0.
4.04/1.97	
4.04/1.97	   Accelerating the following rules:
4.04/1.97	
4.04/1.97	      6: eval1 -> eval1 : A'=0, B'=-A+B, [ A>=1 && B==A ], cost: 2+A
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	   Found no metering function for rule 6.
4.04/1.97	
4.04/1.97	   Removing the simple loops:.
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Accelerated all simple loops using metering functions (where possible):
4.04/1.97	
4.04/1.97	   Start location: start
4.04/1.97	
4.04/1.97	      6: eval1 -> eval1 : A'=0, B'=-A+B, [ A>=1 && B==A ], cost: 2+A
4.04/1.97	
4.04/1.97	      3: start -> eval1 : [], cost: 1
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Chained accelerated rules (with incoming rules):
4.04/1.97	
4.04/1.97	   Start location: start
4.04/1.97	
4.04/1.97	      3: start -> eval1 : [], cost: 1
4.04/1.97	
4.04/1.97	      7: start -> eval1 : A'=0, B'=-A+B, [ A>=1 && B==A ], cost: 3+A
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Removed unreachable locations (and leaf rules with constant cost):
4.04/1.97	
4.04/1.97	   Start location: start
4.04/1.97	
4.04/1.97	      7: start -> eval1 : A'=0, B'=-A+B, [ A>=1 && B==A ], cost: 3+A
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	### Computing asymptotic complexity ###
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Fully simplified ITS problem
4.04/1.97	
4.04/1.97	   Start location: start
4.04/1.97	
4.04/1.97	      7: start -> eval1 : A'=0, B'=-A+B, [ A>=1 && B==A ], cost: 3+A
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Computing asymptotic complexity for rule 7
4.04/1.97	
4.04/1.97	   Solved the limit problem by the following transformations:
4.04/1.97	
4.04/1.97	      Created initial limit problem:
4.04/1.97	
4.04/1.97	      3+A (+), A (+/+!), 1-A+B (+/+!), 1+A-B (+/+!) [not solved]
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	      applying transformation rule (C) using substitution {A==B}
4.04/1.97	
4.04/1.97	      resulting limit problem:
4.04/1.97	
4.04/1.97	      3+B (+), 1 (+/+!), B (+/+!) [not solved]
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	      applying transformation rule (B), deleting 1 (+/+!)
4.04/1.97	
4.04/1.97	      resulting limit problem:
4.04/1.97	
4.04/1.97	      3+B (+), B (+/+!) [not solved]
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	      applying transformation rule (D), replacing 3+B (+) by B (+)
4.04/1.97	
4.04/1.97	      resulting limit problem:
4.04/1.97	
4.04/1.97	      B (+) [solved]
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	   Solution:
4.04/1.97	
4.04/1.97	      A / n
4.04/1.97	
4.04/1.97	      B / n
4.04/1.97	
4.04/1.97	   Resulting cost 3+n has complexity: Poly(n^1)
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Found new complexity Poly(n^1).
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	Obtained the following overall complexity (w.r.t. the length of the input n):
4.04/1.97	
4.04/1.97	   Complexity:  Poly(n^1)
4.04/1.97	
4.04/1.97	   Cpx degree:  1
4.04/1.97	
4.04/1.97	   Solved cost: 3+n
4.04/1.97	
4.04/1.97	   Rule cost:   3+A
4.04/1.97	
4.04/1.97	   Rule guard:  [ A>=1 && B==A ]
4.04/1.97	
4.04/1.97	
4.04/1.97	
4.04/1.97	WORST_CASE(Omega(n^1),?)
4.04/1.97	
4.04/1.97	
4.04/1.97	----------------------------------------
4.04/1.97	
4.04/1.97	(4)
4.04/1.97	BOUNDS(n^1, INF)
4.14/1.99	EOF