4.76/2.91	WORST_CASE(Omega(n^1), O(n^1))
4.76/2.92	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.76/2.92	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.76/2.92	
4.76/2.92	
4.76/2.92	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, nat(2 * Arg_1) + nat(5 * Arg_1) + nat(2 * Arg_0) + nat(6 * Arg_0) + max(1, 1 + 3 * Arg_1)).
4.76/2.92	
4.76/2.92	(0) CpxIntTrs
4.76/2.92	(1) Koat2 Proof [FINISHED, 427 ms]
4.76/2.92	(2) BOUNDS(1, nat(2 * Arg_1) + nat(5 * Arg_1) + nat(2 * Arg_0) + nat(6 * Arg_0) + max(1, 1 + 3 * Arg_1))
4.76/2.92	(3) Loat Proof [FINISHED, 626 ms]
4.76/2.92	(4) BOUNDS(n^1, INF)
4.76/2.92	
4.76/2.92	
4.76/2.92	----------------------------------------
4.76/2.92	
4.76/2.92	(0)
4.76/2.92	Obligation:
4.76/2.92	Complexity Int TRS consisting of the following rules:
4.76/2.92	eval1(A, B) -> Com_1(eval2(A, B)) :|: A >= 1 && B >= 1 && A >= B + 1
4.76/2.92	eval1(A, B) -> Com_1(eval3(A, B)) :|: A >= 1 && B >= 1 && B >= A
4.76/2.92	eval2(A, B) -> Com_1(eval2(A - 1, B)) :|: A >= 1
4.76/2.92	eval2(A, B) -> Com_1(eval1(A, B)) :|: 0 >= A
4.76/2.92	eval3(A, B) -> Com_1(eval3(A, B - 1)) :|: B >= 1
4.76/2.92	eval3(A, B) -> Com_1(eval1(A, B)) :|: 0 >= B
4.76/2.92	start(A, B) -> Com_1(eval1(A, B)) :|: TRUE
4.76/2.92	
4.76/2.92	The start-symbols are:[start_2]
4.76/2.92	
4.76/2.92	
4.76/2.92	----------------------------------------
4.76/2.92	
4.76/2.92	(1) Koat2 Proof (FINISHED)
4.76/2.92	YES( ?, 1+max([0, 3*Arg_1])+max([0, 2*Arg_1])+max([0, 5*Arg_1])+max([0, Arg_0])+max([0, Arg_0])+max([0, 6*Arg_0]) {O(n)})
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Initial Complexity Problem:
4.76/2.92	
4.76/2.92	  Start:  start  
4.76/2.92	
4.76/2.92	  Program_Vars:  Arg_0, Arg_1  
4.76/2.92	
4.76/2.92	  Temp_Vars:    
4.76/2.92	
4.76/2.92	  Locations:  eval1, eval2, eval3, start  
4.76/2.92	
4.76/2.92	  Transitions:
4.76/2.92	
4.76/2.92	  eval1(Arg_0,Arg_1) -> eval2(Arg_0,Arg_1):|:1 <= Arg_0 && 1 <= Arg_1 && Arg_1+1 <= Arg_0
4.76/2.92	
4.76/2.92	  eval1(Arg_0,Arg_1) -> eval3(Arg_0,Arg_1):|:1 <= Arg_0 && 1 <= Arg_1 && Arg_0 <= Arg_1
4.76/2.92	
4.76/2.92	  eval2(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:1 <= Arg_1 && Arg_0 <= 0
4.76/2.92	
4.76/2.92	  eval2(Arg_0,Arg_1) -> eval2(Arg_0-1,Arg_1):|:1 <= Arg_1 && 1 <= Arg_0
4.76/2.92	
4.76/2.92	  eval3(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:1 <= Arg_0 && Arg_1 <= 0
4.76/2.92	
4.76/2.92	  eval3(Arg_0,Arg_1) -> eval3(Arg_0,Arg_1-1):|:1 <= Arg_0 && 1 <= Arg_1
4.76/2.92	
4.76/2.92	  start(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:  
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Timebounds: 
4.76/2.92	
4.76/2.92	  Overall timebound: 1+max([0, 3*Arg_1])+max([0, 2*Arg_1])+max([0, 5*Arg_1])+max([0, Arg_0])+max([0, Arg_0])+max([0, 6*Arg_0]) {O(n)}
4.76/2.92	
4.76/2.92	  0: eval1->eval2: max([0, Arg_0]) {O(n)}
4.76/2.92	
4.76/2.92	  1: eval1->eval3: max([0, 2*Arg_1]) {O(n)}
4.76/2.92	
4.76/2.92	  2: eval2->eval2: max([0, Arg_0]) {O(n)}
4.76/2.92	
4.76/2.92	  3: eval2->eval1: max([0, 6*Arg_0]) {O(n)}
4.76/2.92	
4.76/2.92	  4: eval3->eval3: max([0, 3*Arg_1]) {O(n)}
4.76/2.92	
4.76/2.92	  5: eval3->eval1: max([0, 5*Arg_1]) {O(n)}
4.76/2.92	
4.76/2.92	  6: start->eval1: 1 {O(1)}
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Costbounds:
4.76/2.92	
4.76/2.92	  Overall costbound: 1+max([0, 3*Arg_1])+max([0, 2*Arg_1])+max([0, 5*Arg_1])+max([0, Arg_0])+max([0, Arg_0])+max([0, 6*Arg_0]) {O(n)}
4.76/2.92	
4.76/2.92	  0: eval1->eval2: max([0, Arg_0]) {O(n)}
4.76/2.92	
4.76/2.92	  1: eval1->eval3: max([0, 2*Arg_1]) {O(n)}
4.76/2.92	
4.76/2.92	  2: eval2->eval2: max([0, Arg_0]) {O(n)}
4.76/2.92	
4.76/2.92	  3: eval2->eval1: max([0, 6*Arg_0]) {O(n)}
4.76/2.92	
4.76/2.92	  4: eval3->eval3: max([0, 3*Arg_1]) {O(n)}
4.76/2.92	
4.76/2.92	  5: eval3->eval1: max([0, 5*Arg_1]) {O(n)}
4.76/2.92	
4.76/2.92	  6: start->eval1: 1 {O(1)}
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Sizebounds:
4.76/2.92	
4.76/2.92	`Lower:
4.76/2.92	
4.76/2.92	  0: eval1->eval2, Arg_0: 2 {O(1)}
4.76/2.92	
4.76/2.92	  0: eval1->eval2, Arg_1: 1 {O(1)}
4.76/2.92	
4.76/2.92	  1: eval1->eval3, Arg_0: 1 {O(1)}
4.76/2.92	
4.76/2.92	  1: eval1->eval3, Arg_1: 1 {O(1)}
4.76/2.92	
4.76/2.92	  2: eval2->eval2, Arg_0: 0 {O(1)}
4.76/2.92	
4.76/2.92	  2: eval2->eval2, Arg_1: 1 {O(1)}
4.76/2.92	
4.76/2.92	  3: eval2->eval1, Arg_0: 0 {O(1)}
4.76/2.92	
4.76/2.92	  3: eval2->eval1, Arg_1: 1 {O(1)}
4.76/2.92	
4.76/2.92	  4: eval3->eval3, Arg_0: 1 {O(1)}
4.76/2.92	
4.76/2.92	  4: eval3->eval3, Arg_1: 0 {O(1)}
4.76/2.92	
4.76/2.92	  5: eval3->eval1, Arg_0: 1 {O(1)}
4.76/2.92	
4.76/2.92	  5: eval3->eval1, Arg_1: 0 {O(1)}
4.76/2.92	
4.76/2.92	  6: start->eval1, Arg_0: Arg_0 {O(n)}
4.76/2.92	
4.76/2.92	  6: start->eval1, Arg_1: Arg_1 {O(n)}
4.76/2.92	
4.76/2.92	`Upper:
4.76/2.92	
4.76/2.92	  0: eval1->eval2, Arg_0: Arg_0 {O(n)}
4.76/2.92	
4.76/2.92	  0: eval1->eval2, Arg_1: Arg_1 {O(n)}
4.76/2.92	
4.76/2.92	  1: eval1->eval3, Arg_0: Arg_0 {O(n)}
4.76/2.92	
4.76/2.92	  1: eval1->eval3, Arg_1: Arg_1 {O(n)}
4.76/2.92	
4.76/2.92	  2: eval2->eval2, Arg_0: Arg_0 {O(n)}
4.76/2.92	
4.76/2.92	  2: eval2->eval2, Arg_1: Arg_1 {O(n)}
4.76/2.92	
4.76/2.92	  3: eval2->eval1, Arg_0: 0 {O(1)}
4.76/2.92	
4.76/2.92	  3: eval2->eval1, Arg_1: Arg_1 {O(n)}
4.76/2.92	
4.76/2.92	  4: eval3->eval3, Arg_0: Arg_0 {O(n)}
4.76/2.92	
4.76/2.92	  4: eval3->eval3, Arg_1: Arg_1 {O(n)}
4.76/2.92	
4.76/2.92	  5: eval3->eval1, Arg_0: Arg_0 {O(n)}
4.76/2.92	
4.76/2.92	  5: eval3->eval1, Arg_1: 0 {O(1)}
4.76/2.92	
4.76/2.92	  6: start->eval1, Arg_0: Arg_0 {O(n)}
4.76/2.92	
4.76/2.92	  6: start->eval1, Arg_1: Arg_1 {O(n)}
4.76/2.92	
4.76/2.92	
4.76/2.92	----------------------------------------
4.76/2.92	
4.76/2.92	(2)
4.76/2.92	BOUNDS(1, nat(2 * Arg_1) + nat(5 * Arg_1) + nat(2 * Arg_0) + nat(6 * Arg_0) + max(1, 1 + 3 * Arg_1))
4.76/2.92	
4.76/2.92	----------------------------------------
4.76/2.92	
4.76/2.92	(3) Loat Proof (FINISHED)
4.76/2.92	
4.76/2.92	
4.76/2.92	### Pre-processing the ITS problem ###
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Initial linear ITS problem
4.76/2.92	
4.76/2.92	   Start location: start
4.76/2.92	
4.76/2.92	      0: eval1 -> eval2 : [ A>=1 && B>=1 && A>=1+B ], cost: 1
4.76/2.92	
4.76/2.92	      1: eval1 -> eval3 : [ A>=1 && B>=1 && B>=A ], cost: 1
4.76/2.92	
4.76/2.92	      2: eval2 -> eval2 : A'=-1+A, [ A>=1 ], cost: 1
4.76/2.92	
4.76/2.92	      3: eval2 -> eval1 : [ 0>=A ], cost: 1
4.76/2.92	
4.76/2.92	      4: eval3 -> eval3 : B'=-1+B, [ B>=1 ], cost: 1
4.76/2.92	
4.76/2.92	      5: eval3 -> eval1 : [ 0>=B ], cost: 1
4.76/2.92	
4.76/2.92	      6: start -> eval1 : [], cost: 1
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Simplified all rules, resulting in:
4.76/2.92	
4.76/2.92	   Start location: start
4.76/2.92	
4.76/2.92	      0: eval1 -> eval2 : [ B>=1 && A>=1+B ], cost: 1
4.76/2.92	
4.76/2.92	      1: eval1 -> eval3 : [ A>=1 && B>=1 && B>=A ], cost: 1
4.76/2.92	
4.76/2.92	      2: eval2 -> eval2 : A'=-1+A, [ A>=1 ], cost: 1
4.76/2.92	
4.76/2.92	      3: eval2 -> eval1 : [ 0>=A ], cost: 1
4.76/2.92	
4.76/2.92	      4: eval3 -> eval3 : B'=-1+B, [ B>=1 ], cost: 1
4.76/2.92	
4.76/2.92	      5: eval3 -> eval1 : [ 0>=B ], cost: 1
4.76/2.92	
4.76/2.92	      6: start -> eval1 : [], cost: 1
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	### Simplification by acceleration and chaining ###
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Accelerating simple loops of location 1.
4.76/2.92	
4.76/2.92	   Accelerating the following rules:
4.76/2.92	
4.76/2.92	      2: eval2 -> eval2 : A'=-1+A, [ A>=1 ], cost: 1
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	   Accelerated rule 2 with metering function A, yielding the new rule 7.
4.76/2.92	
4.76/2.92	   Removing the simple loops: 2.
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Accelerating simple loops of location 2.
4.76/2.92	
4.76/2.92	   Accelerating the following rules:
4.76/2.92	
4.76/2.92	      4: eval3 -> eval3 : B'=-1+B, [ B>=1 ], cost: 1
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	   Accelerated rule 4 with metering function B, yielding the new rule 8.
4.76/2.92	
4.76/2.92	   Removing the simple loops: 4.
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Accelerated all simple loops using metering functions (where possible):
4.76/2.92	
4.76/2.92	   Start location: start
4.76/2.92	
4.76/2.92	      0: eval1 -> eval2 : [ B>=1 && A>=1+B ], cost: 1
4.76/2.92	
4.76/2.92	      1: eval1 -> eval3 : [ A>=1 && B>=1 && B>=A ], cost: 1
4.76/2.92	
4.76/2.92	      3: eval2 -> eval1 : [ 0>=A ], cost: 1
4.76/2.92	
4.76/2.92	      7: eval2 -> eval2 : A'=0, [ A>=1 ], cost: A
4.76/2.92	
4.76/2.92	      5: eval3 -> eval1 : [ 0>=B ], cost: 1
4.76/2.92	
4.76/2.92	      8: eval3 -> eval3 : B'=0, [ B>=1 ], cost: B
4.76/2.92	
4.76/2.92	      6: start -> eval1 : [], cost: 1
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Chained accelerated rules (with incoming rules):
4.76/2.92	
4.76/2.92	   Start location: start
4.76/2.92	
4.76/2.92	      0: eval1 -> eval2 : [ B>=1 && A>=1+B ], cost: 1
4.76/2.92	
4.76/2.92	      1: eval1 -> eval3 : [ A>=1 && B>=1 && B>=A ], cost: 1
4.76/2.92	
4.76/2.92	      9: eval1 -> eval2 : A'=0, [ B>=1 && A>=1+B && A>=1 ], cost: 1+A
4.76/2.92	
4.76/2.92	     10: eval1 -> eval3 : B'=0, [ A>=1 && B>=1 && B>=A ], cost: 1+B
4.76/2.92	
4.76/2.92	      3: eval2 -> eval1 : [ 0>=A ], cost: 1
4.76/2.92	
4.76/2.92	      5: eval3 -> eval1 : [ 0>=B ], cost: 1
4.76/2.92	
4.76/2.92	      6: start -> eval1 : [], cost: 1
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Eliminated locations (on tree-shaped paths):
4.76/2.92	
4.76/2.92	   Start location: start
4.76/2.92	
4.76/2.92	     11: eval1 -> eval1 : A'=0, [ B>=1 && A>=1+B && A>=1 ], cost: 2+A
4.76/2.92	
4.76/2.92	     12: eval1 -> eval1 : B'=0, [ A>=1 && B>=1 && B>=A ], cost: 2+B
4.76/2.92	
4.76/2.92	      6: start -> eval1 : [], cost: 1
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Accelerating simple loops of location 0.
4.76/2.92	
4.76/2.92	   Accelerating the following rules:
4.76/2.92	
4.76/2.92	     11: eval1 -> eval1 : A'=0, [ B>=1 && A>=1+B && A>=1 ], cost: 2+A
4.76/2.92	
4.76/2.92	     12: eval1 -> eval1 : B'=0, [ A>=1 && B>=1 && B>=A ], cost: 2+B
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	   Accelerated rule 11 with NONTERM (after strengthening guard), yielding the new rule 13.
4.76/2.92	
4.76/2.92	   Accelerated rule 12 with NONTERM (after strengthening guard), yielding the new rule 14.
4.76/2.92	
4.76/2.92	   Removing the simple loops:.
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Accelerated all simple loops using metering functions (where possible):
4.76/2.92	
4.76/2.92	   Start location: start
4.76/2.92	
4.76/2.92	     11: eval1 -> eval1 : A'=0, [ B>=1 && A>=1+B && A>=1 ], cost: 2+A
4.76/2.92	
4.76/2.92	     12: eval1 -> eval1 : B'=0, [ A>=1 && B>=1 && B>=A ], cost: 2+B
4.76/2.92	
4.76/2.92	     13: eval1 -> [6] : [ B>=1 && A>=1+B && A>=1 && 0>=1+B ], cost: INF
4.76/2.92	
4.76/2.92	     14: eval1 -> [6] : [ A>=1 && B>=1 && B>=A && 0>=A ], cost: INF
4.76/2.92	
4.76/2.92	      6: start -> eval1 : [], cost: 1
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Chained accelerated rules (with incoming rules):
4.76/2.92	
4.76/2.92	   Start location: start
4.76/2.92	
4.76/2.92	      6: start -> eval1 : [], cost: 1
4.76/2.92	
4.76/2.92	     15: start -> eval1 : A'=0, [ B>=1 && A>=1+B && A>=1 ], cost: 3+A
4.76/2.92	
4.76/2.92	     16: start -> eval1 : B'=0, [ A>=1 && B>=1 && B>=A ], cost: 3+B
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Removed unreachable locations (and leaf rules with constant cost):
4.76/2.92	
4.76/2.92	   Start location: start
4.76/2.92	
4.76/2.92	     15: start -> eval1 : A'=0, [ B>=1 && A>=1+B && A>=1 ], cost: 3+A
4.76/2.92	
4.76/2.92	     16: start -> eval1 : B'=0, [ A>=1 && B>=1 && B>=A ], cost: 3+B
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	### Computing asymptotic complexity ###
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Fully simplified ITS problem
4.76/2.92	
4.76/2.92	   Start location: start
4.76/2.92	
4.76/2.92	     15: start -> eval1 : A'=0, [ B>=1 && A>=1+B && A>=1 ], cost: 3+A
4.76/2.92	
4.76/2.92	     16: start -> eval1 : B'=0, [ A>=1 && B>=1 && B>=A ], cost: 3+B
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Computing asymptotic complexity for rule 15
4.76/2.92	
4.76/2.92	   Solved the limit problem by the following transformations:
4.76/2.92	
4.76/2.92	      Created initial limit problem:
4.76/2.92	
4.76/2.92	      3+A (+), A-B (+/+!), B (+/+!) [not solved]
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	      applying transformation rule (C) using substitution {B==1}
4.76/2.92	
4.76/2.92	      resulting limit problem:
4.76/2.92	
4.76/2.92	      1 (+/+!), -1+A (+/+!), 3+A (+) [not solved]
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	      applying transformation rule (C) using substitution {A==1+B}
4.76/2.92	
4.76/2.92	      resulting limit problem:
4.76/2.92	
4.76/2.92	      1 (+/+!), B (+/+!), 4+B (+) [not solved]
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	      applying transformation rule (B), deleting 1 (+/+!)
4.76/2.92	
4.76/2.92	      resulting limit problem:
4.76/2.92	
4.76/2.92	      B (+/+!), 4+B (+) [not solved]
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	      applying transformation rule (D), replacing 4+B (+) by B (+)
4.76/2.92	
4.76/2.92	      resulting limit problem:
4.76/2.92	
4.76/2.92	      B (+) [solved]
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	   Solution:
4.76/2.92	
4.76/2.92	      A / 1+n
4.76/2.92	
4.76/2.92	      B / 1
4.76/2.92	
4.76/2.92	   Resulting cost 4+n has complexity: Poly(n^1)
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Found new complexity Poly(n^1).
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	Obtained the following overall complexity (w.r.t. the length of the input n):
4.76/2.92	
4.76/2.92	   Complexity:  Poly(n^1)
4.76/2.92	
4.76/2.92	   Cpx degree:  1
4.76/2.92	
4.76/2.92	   Solved cost: 4+n
4.76/2.92	
4.76/2.92	   Rule cost:   3+A
4.76/2.92	
4.76/2.92	   Rule guard:  [ B>=1 && A>=1+B ]
4.76/2.92	
4.76/2.92	
4.76/2.92	
4.76/2.92	WORST_CASE(Omega(n^1),?)
4.76/2.92	
4.76/2.92	
4.76/2.92	----------------------------------------
4.76/2.92	
4.76/2.92	(4)
4.76/2.92	BOUNDS(n^1, INF)
4.82/2.94	EOF