4.27/2.11	WORST_CASE(Omega(n^2), O(n^2))
4.27/2.11	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.27/2.11	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.27/2.11	
4.27/2.11	
4.27/2.11	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, max(1, 3 + 2 * Arg_0) + nat(3 * Arg_0) + nat(Arg_0 * max(3 * Arg_0, -3) + max(3 * Arg_0, -3))).
4.27/2.11	
4.27/2.11	(0) CpxIntTrs
4.27/2.11	(1) Koat2 Proof [FINISHED, 249 ms]
4.27/2.11	(2) BOUNDS(1, max(1, 3 + 2 * Arg_0) + nat(3 * Arg_0) + nat(Arg_0 * max(3 * Arg_0, -3) + max(3 * Arg_0, -3)))
4.27/2.11	(3) Loat Proof [FINISHED, 416 ms]
4.27/2.11	(4) BOUNDS(n^2, INF)
4.27/2.11	
4.27/2.11	
4.27/2.11	----------------------------------------
4.27/2.11	
4.27/2.11	(0)
4.27/2.11	Obligation:
4.27/2.11	Complexity Int TRS consisting of the following rules:
4.27/2.11	eval1(A, B) -> Com_1(eval2(A, 0)) :|: A >= 0
4.27/2.11	eval2(A, B) -> Com_1(eval2(A, B + 1)) :|: A >= 1 + B
4.27/2.11	eval2(A, B) -> Com_1(eval1(A - 1, B)) :|: B >= A
4.27/2.11	start(A, B) -> Com_1(eval1(A, B)) :|: TRUE
4.27/2.11	
4.27/2.11	The start-symbols are:[start_2]
4.27/2.11	
4.27/2.11	
4.27/2.11	----------------------------------------
4.27/2.11	
4.27/2.11	(1) Koat2 Proof (FINISHED)
4.27/2.11	YES( ?, 1+2*max([0, 1+Arg_0])+max([0, 3*Arg_0])+max([0, (1+Arg_0)*max([-3, 3*Arg_0])]) {O(n^2)})
4.27/2.11	
4.27/2.11	
4.27/2.11	
4.27/2.11	Initial Complexity Problem:
4.27/2.11	
4.27/2.11	  Start:  start  
4.27/2.11	
4.27/2.11	  Program_Vars:  Arg_0, Arg_1  
4.27/2.11	
4.27/2.11	  Temp_Vars:    
4.27/2.11	
4.27/2.11	  Locations:  eval1, eval2, start  
4.27/2.11	
4.27/2.11	  Transitions:
4.27/2.11	
4.27/2.11	  eval1(Arg_0,Arg_1) -> eval2(Arg_0,0):|:0 <= Arg_0
4.27/2.11	
4.27/2.11	  eval2(Arg_0,Arg_1) -> eval1(Arg_0-1,Arg_1):|:Arg_1 <= Arg_0 && 0 <= Arg_1 && 0 <= Arg_0+Arg_1 && 0 <= Arg_0 && Arg_0 <= Arg_1
4.27/2.11	
4.27/2.11	  eval2(Arg_0,Arg_1) -> eval2(Arg_0,Arg_1+1):|:Arg_1 <= Arg_0 && 0 <= Arg_1 && 0 <= Arg_0+Arg_1 && 0 <= Arg_0 && 1+Arg_1 <= Arg_0
4.27/2.11	
4.27/2.11	  start(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:  
4.27/2.11	
4.27/2.11	
4.27/2.11	
4.27/2.11	Timebounds: 
4.27/2.11	
4.27/2.11	  Overall timebound: 1+2*max([0, 1+Arg_0])+max([0, 3*Arg_0])+max([0, (1+Arg_0)*max([-3, 3*Arg_0])]) {O(n^2)}
4.27/2.11	
4.27/2.11	  0: eval1->eval2: max([0, 1+Arg_0]) {O(n)}
4.27/2.11	
4.27/2.11	  1: eval2->eval2: max([0, 3*Arg_0])+max([0, (1+Arg_0)*max([-3, 3*Arg_0])]) {O(n^2)}
4.27/2.11	
4.27/2.11	  2: eval2->eval1: max([0, 1+Arg_0]) {O(n)}
4.27/2.11	
4.27/2.11	  3: start->eval1: 1 {O(1)}
4.27/2.11	
4.27/2.11	
4.27/2.11	
4.27/2.11	Costbounds:
4.27/2.11	
4.27/2.11	  Overall costbound: 1+2*max([0, 1+Arg_0])+max([0, 3*Arg_0])+max([0, (1+Arg_0)*max([-3, 3*Arg_0])]) {O(n^2)}
4.27/2.11	
4.27/2.11	  0: eval1->eval2: max([0, 1+Arg_0]) {O(n)}
4.27/2.11	
4.27/2.11	  1: eval2->eval2: max([0, 3*Arg_0])+max([0, (1+Arg_0)*max([-3, 3*Arg_0])]) {O(n^2)}
4.27/2.11	
4.27/2.11	  2: eval2->eval1: max([0, 1+Arg_0]) {O(n)}
4.27/2.11	
4.27/2.11	  3: start->eval1: 1 {O(1)}
4.27/2.11	
4.27/2.11	
4.27/2.11	
4.27/2.11	Sizebounds:
4.27/2.11	
4.27/2.11	`Lower:
4.27/2.11	
4.27/2.11	  0: eval1->eval2, Arg_0: 0 {O(1)}
4.27/2.11	
4.27/2.11	  0: eval1->eval2, Arg_1: 0 {O(1)}
4.27/2.11	
4.27/2.11	  1: eval2->eval2, Arg_0: 1 {O(1)}
4.27/2.11	
4.27/2.11	  1: eval2->eval2, Arg_1: 1 {O(1)}
4.27/2.11	
4.27/2.11	  2: eval2->eval1, Arg_0: -1 {O(1)}
4.27/2.11	
4.27/2.11	  2: eval2->eval1, Arg_1: 0 {O(1)}
4.27/2.11	
4.27/2.11	  3: start->eval1, Arg_0: Arg_0 {O(n)}
4.27/2.11	
4.27/2.11	  3: start->eval1, Arg_1: Arg_1 {O(n)}
4.27/2.11	
4.27/2.11	`Upper:
4.27/2.11	
4.27/2.11	  0: eval1->eval2, Arg_0: Arg_0 {O(n)}
4.27/2.11	
4.27/2.11	  0: eval1->eval2, Arg_1: 0 {O(1)}
4.27/2.11	
4.27/2.11	  1: eval2->eval2, Arg_0: Arg_0 {O(n)}
4.27/2.11	
4.27/2.11	  1: eval2->eval2, Arg_1: max([0, 3*Arg_0])+max([0, (1+Arg_0)*max([-3, 3*Arg_0])]) {O(n^2)}
4.27/2.11	
4.27/2.11	  2: eval2->eval1, Arg_0: Arg_0 {O(n)}
4.27/2.11	
4.27/2.11	  2: eval2->eval1, Arg_1: max([0, max([0, 3*Arg_0])+max([0, (1+Arg_0)*max([-3, 3*Arg_0])])]) {O(n^2)}
4.27/2.11	
4.27/2.11	  3: start->eval1, Arg_0: Arg_0 {O(n)}
4.27/2.11	
4.27/2.11	  3: start->eval1, Arg_1: Arg_1 {O(n)}
4.27/2.11	
4.27/2.11	
4.27/2.11	----------------------------------------
4.27/2.11	
4.27/2.11	(2)
4.27/2.11	BOUNDS(1, max(1, 3 + 2 * Arg_0) + nat(3 * Arg_0) + nat(Arg_0 * max(3 * Arg_0, -3) + max(3 * Arg_0, -3)))
4.27/2.11	
4.27/2.11	----------------------------------------
4.27/2.11	
4.27/2.11	(3) Loat Proof (FINISHED)
4.27/2.11	
4.27/2.11	
4.27/2.11	### Pre-processing the ITS problem ###
4.27/2.11	
4.27/2.11	
4.27/2.11	
4.27/2.11	Initial linear ITS problem
4.27/2.11	
4.27/2.11	   Start location: start
4.27/2.11	
4.27/2.11	      0: eval1 -> eval2 : B'=0, [ A>=0 ], cost: 1
4.27/2.11	
4.27/2.11	      1: eval2 -> eval2 : B'=1+B, [ A>=1+B ], cost: 1
4.27/2.11	
4.27/2.11	      2: eval2 -> eval1 : A'=-1+A, [ B>=A ], cost: 1
4.27/2.11	
4.27/2.11	      3: start -> eval1 : [], cost: 1
4.27/2.11	
4.27/2.11	
4.27/2.11	
4.27/2.11	### Simplification by acceleration and chaining ###
4.27/2.11	
4.27/2.11	
4.27/2.11	
4.27/2.11	Accelerating simple loops of location 1.
4.27/2.11	
4.27/2.11	   Accelerating the following rules:
4.27/2.11	
4.27/2.11	      1: eval2 -> eval2 : B'=1+B, [ A>=1+B ], cost: 1
4.27/2.11	
4.27/2.11	
4.27/2.11	
4.27/2.11	   Accelerated rule 1 with metering function A-B, yielding the new rule 4.
4.27/2.11	
4.27/2.11	   Removing the simple loops: 1.
4.27/2.11	
4.27/2.11	
4.27/2.11	
4.27/2.11	Accelerated all simple loops using metering functions (where possible):
4.27/2.11	
4.27/2.11	   Start location: start
4.27/2.11	
4.27/2.11	      0: eval1 -> eval2 : B'=0, [ A>=0 ], cost: 1
4.27/2.11	
4.27/2.11	      2: eval2 -> eval1 : A'=-1+A, [ B>=A ], cost: 1
4.27/2.11	
4.27/2.11	      4: eval2 -> eval2 : B'=A, [ A>=1+B ], cost: A-B
4.27/2.11	
4.27/2.11	      3: start -> eval1 : [], cost: 1
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	Chained accelerated rules (with incoming rules):
4.27/2.12	
4.27/2.12	   Start location: start
4.27/2.12	
4.27/2.12	      0: eval1 -> eval2 : B'=0, [ A>=0 ], cost: 1
4.27/2.12	
4.27/2.12	      5: eval1 -> eval2 : B'=A, [ A>=1 ], cost: 1+A
4.27/2.12	
4.27/2.12	      2: eval2 -> eval1 : A'=-1+A, [ B>=A ], cost: 1
4.27/2.12	
4.27/2.12	      3: start -> eval1 : [], cost: 1
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	Eliminated locations (on tree-shaped paths):
4.27/2.12	
4.27/2.12	   Start location: start
4.27/2.12	
4.27/2.12	      6: eval1 -> eval1 : A'=-1+A, B'=0, [ A>=0 && 0>=A ], cost: 2
4.27/2.12	
4.27/2.12	      7: eval1 -> eval1 : A'=-1+A, B'=A, [ A>=1 ], cost: 2+A
4.27/2.12	
4.27/2.12	      3: start -> eval1 : [], cost: 1
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	Accelerating simple loops of location 0.
4.27/2.12	
4.27/2.12	   Simplified some of the simple loops (and removed duplicate rules).
4.27/2.12	
4.27/2.12	   Accelerating the following rules:
4.27/2.12	
4.27/2.12	      6: eval1 -> eval1 : A'=-1+A, B'=0, [ -A==0 ], cost: 2
4.27/2.12	
4.27/2.12	      7: eval1 -> eval1 : A'=-1+A, B'=A, [ A>=1 ], cost: 2+A
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	   Accelerated rule 6 with metering function 1+A, yielding the new rule 8.
4.27/2.12	
4.27/2.12	   Accelerated rule 7 with metering function A, yielding the new rule 9.
4.27/2.12	
4.27/2.12	   Removing the simple loops: 6 7.
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	Accelerated all simple loops using metering functions (where possible):
4.27/2.12	
4.27/2.12	   Start location: start
4.27/2.12	
4.27/2.12	      8: eval1 -> eval1 : A'=-1, B'=0, [ -A==0 ], cost: 2+2*A
4.27/2.12	
4.27/2.12	      9: eval1 -> eval1 : A'=0, B'=1, [ A>=1 ], cost: 5/2*A+1/2*A^2
4.27/2.12	
4.27/2.12	      3: start -> eval1 : [], cost: 1
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	Chained accelerated rules (with incoming rules):
4.27/2.12	
4.27/2.12	   Start location: start
4.27/2.12	
4.27/2.12	      3: start -> eval1 : [], cost: 1
4.27/2.12	
4.27/2.12	     10: start -> eval1 : A'=-1, B'=0, [ -A==0 ], cost: 3+2*A
4.27/2.12	
4.27/2.12	     11: start -> eval1 : A'=0, B'=1, [ A>=1 ], cost: 1+5/2*A+1/2*A^2
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	Removed unreachable locations (and leaf rules with constant cost):
4.27/2.12	
4.27/2.12	   Start location: start
4.27/2.12	
4.27/2.12	     10: start -> eval1 : A'=-1, B'=0, [ -A==0 ], cost: 3+2*A
4.27/2.12	
4.27/2.12	     11: start -> eval1 : A'=0, B'=1, [ A>=1 ], cost: 1+5/2*A+1/2*A^2
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	### Computing asymptotic complexity ###
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	Fully simplified ITS problem
4.27/2.12	
4.27/2.12	   Start location: start
4.27/2.12	
4.27/2.12	     10: start -> eval1 : A'=-1, B'=0, [ -A==0 ], cost: 3+2*A
4.27/2.12	
4.27/2.12	     11: start -> eval1 : A'=0, B'=1, [ A>=1 ], cost: 1+5/2*A+1/2*A^2
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	Computing asymptotic complexity for rule 10
4.27/2.12	
4.27/2.12	   Could not solve the limit problem.
4.27/2.12	
4.27/2.12	   Resulting cost 0 has complexity: Unknown
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	Computing asymptotic complexity for rule 11
4.27/2.12	
4.27/2.12	   Solved the limit problem by the following transformations:
4.27/2.12	
4.27/2.12	      Created initial limit problem:
4.27/2.12	
4.27/2.12	      1+5/2*A+1/2*A^2 (+), A (+/+!) [not solved]
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	      removing all constraints (solved by SMT)
4.27/2.12	
4.27/2.12	      resulting limit problem: [solved]
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	      applying transformation rule (C) using substitution {A==n}
4.27/2.12	
4.27/2.12	      resulting limit problem:
4.27/2.12	
4.27/2.12	      [solved]
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	   Solution:
4.27/2.12	
4.27/2.12	      A / n
4.27/2.12	
4.27/2.12	   Resulting cost 1+1/2*n^2+5/2*n has complexity: Poly(n^2)
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	Found new complexity Poly(n^2).
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	Obtained the following overall complexity (w.r.t. the length of the input n):
4.27/2.12	
4.27/2.12	   Complexity:  Poly(n^2)
4.27/2.12	
4.27/2.12	   Cpx degree:  2
4.27/2.12	
4.27/2.12	   Solved cost: 1+1/2*n^2+5/2*n
4.27/2.12	
4.27/2.12	   Rule cost:   1+5/2*A+1/2*A^2
4.27/2.12	
4.27/2.12	   Rule guard:  [ A>=1 ]
4.27/2.12	
4.27/2.12	
4.27/2.12	
4.27/2.12	WORST_CASE(Omega(n^2),?)
4.27/2.12	
4.27/2.12	
4.27/2.12	----------------------------------------
4.27/2.12	
4.27/2.12	(4)
4.27/2.12	BOUNDS(n^2, INF)
4.27/2.15	EOF