3.74/1.85	WORST_CASE(Omega(n^1), O(n^1))
3.74/1.86	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.74/1.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.74/1.86	
3.74/1.86	
3.74/1.86	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(3, 2 + Arg_0) + max(1, 1 + Arg_0)).
3.74/1.86	
3.74/1.86	(0) CpxIntTrs
3.74/1.86	(1) Koat2 Proof [FINISHED, 133 ms]
3.74/1.86	(2) BOUNDS(1, max(3, 2 + Arg_0) + max(1, 1 + Arg_0))
3.74/1.86	(3) Loat Proof [FINISHED, 215 ms]
3.74/1.86	(4) BOUNDS(n^1, INF)
3.74/1.86	
3.74/1.86	
3.74/1.86	----------------------------------------
3.74/1.86	
3.74/1.86	(0)
3.74/1.86	Obligation:
3.74/1.86	Complexity Int TRS consisting of the following rules:
3.74/1.86	evalndecrstart(A) -> Com_1(evalndecrentryin(A)) :|: TRUE
3.74/1.86	evalndecrentryin(A) -> Com_1(evalndecrbb1in(A - 1)) :|: TRUE
3.74/1.86	evalndecrbb1in(A) -> Com_1(evalndecrbbin(A)) :|: A >= 2
3.74/1.86	evalndecrbb1in(A) -> Com_1(evalndecrreturnin(A)) :|: 1 >= A
3.74/1.86	evalndecrbbin(A) -> Com_1(evalndecrbb1in(A - 1)) :|: TRUE
3.74/1.86	evalndecrreturnin(A) -> Com_1(evalndecrstop(A)) :|: TRUE
3.74/1.86	
3.74/1.86	The start-symbols are:[evalndecrstart_1]
3.74/1.86	
3.74/1.86	
3.74/1.86	----------------------------------------
3.74/1.86	
3.74/1.86	(1) Koat2 Proof (FINISHED)
3.74/1.86	YES( ?, 3+max([0, -1+Arg_0])+max([1, 1+Arg_0]) {O(n)})
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Initial Complexity Problem:
3.74/1.86	
3.74/1.86	  Start:  evalndecrstart  
3.74/1.86	
3.74/1.86	  Program_Vars:  Arg_0  
3.74/1.86	
3.74/1.86	  Temp_Vars:    
3.74/1.86	
3.74/1.86	  Locations:  evalndecrbb1in, evalndecrbbin, evalndecrentryin, evalndecrreturnin, evalndecrstart, evalndecrstop  
3.74/1.86	
3.74/1.86	  Transitions:
3.74/1.86	
3.74/1.86	  evalndecrbb1in(Arg_0) -> evalndecrbbin(Arg_0):|:2 <= Arg_0
3.74/1.86	
3.74/1.86	  evalndecrbb1in(Arg_0) -> evalndecrreturnin(Arg_0):|:Arg_0 <= 1
3.74/1.86	
3.74/1.86	  evalndecrbbin(Arg_0) -> evalndecrbb1in(Arg_0-1):|:2 <= Arg_0
3.74/1.86	
3.74/1.86	  evalndecrentryin(Arg_0) -> evalndecrbb1in(Arg_0-1):|:
3.74/1.86	
3.74/1.86	  evalndecrreturnin(Arg_0) -> evalndecrstop(Arg_0):|:Arg_0 <= 1
3.74/1.86	
3.74/1.86	  evalndecrstart(Arg_0) -> evalndecrentryin(Arg_0):|:  
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Timebounds: 
3.74/1.86	
3.74/1.86	  Overall timebound: 3+max([0, -1+Arg_0])+max([1, 1+Arg_0]) {O(n)}
3.74/1.86	
3.74/1.86	  2: evalndecrbb1in->evalndecrbbin: max([0, Arg_0]) {O(n)}
3.74/1.86	
3.74/1.86	  3: evalndecrbb1in->evalndecrreturnin: 1 {O(1)}
3.74/1.86	
3.74/1.86	  4: evalndecrbbin->evalndecrbb1in: max([0, -1+Arg_0]) {O(n)}
3.74/1.86	
3.74/1.86	  1: evalndecrentryin->evalndecrbb1in: 1 {O(1)}
3.74/1.86	
3.74/1.86	  5: evalndecrreturnin->evalndecrstop: 1 {O(1)}
3.74/1.86	
3.74/1.86	  0: evalndecrstart->evalndecrentryin: 1 {O(1)}
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Costbounds:
3.74/1.86	
3.74/1.86	  Overall costbound: 3+max([0, -1+Arg_0])+max([1, 1+Arg_0]) {O(n)}
3.74/1.86	
3.74/1.86	  2: evalndecrbb1in->evalndecrbbin: max([0, Arg_0]) {O(n)}
3.74/1.86	
3.74/1.86	  3: evalndecrbb1in->evalndecrreturnin: 1 {O(1)}
3.74/1.86	
3.74/1.86	  4: evalndecrbbin->evalndecrbb1in: max([0, -1+Arg_0]) {O(n)}
3.74/1.86	
3.74/1.86	  1: evalndecrentryin->evalndecrbb1in: 1 {O(1)}
3.74/1.86	
3.74/1.86	  5: evalndecrreturnin->evalndecrstop: 1 {O(1)}
3.74/1.86	
3.74/1.86	  0: evalndecrstart->evalndecrentryin: 1 {O(1)}
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Sizebounds:
3.74/1.86	
3.74/1.86	`Lower:
3.74/1.86	
3.74/1.86	  2: evalndecrbb1in->evalndecrbbin, Arg_0: 2 {O(1)}
3.74/1.86	
3.74/1.86	  3: evalndecrbb1in->evalndecrreturnin, Arg_0: min([1, -(1-Arg_0)]) {O(n)}
3.74/1.86	
3.74/1.86	  4: evalndecrbbin->evalndecrbb1in, Arg_0: 1 {O(1)}
3.74/1.86	
3.74/1.86	  1: evalndecrentryin->evalndecrbb1in, Arg_0: -1+Arg_0 {O(n)}
3.74/1.86	
3.74/1.86	  5: evalndecrreturnin->evalndecrstop, Arg_0: min([1, -(1-Arg_0)]) {O(n)}
3.74/1.86	
3.74/1.86	  0: evalndecrstart->evalndecrentryin, Arg_0: Arg_0 {O(n)}
3.74/1.86	
3.74/1.86	`Upper:
3.74/1.86	
3.74/1.86	  2: evalndecrbb1in->evalndecrbbin, Arg_0: -1+Arg_0 {O(n)}
3.74/1.86	
3.74/1.86	  3: evalndecrbb1in->evalndecrreturnin, Arg_0: 1 {O(1)}
3.74/1.86	
3.74/1.86	  4: evalndecrbbin->evalndecrbb1in, Arg_0: -1+Arg_0 {O(n)}
3.74/1.86	
3.74/1.86	  1: evalndecrentryin->evalndecrbb1in, Arg_0: -1+Arg_0 {O(n)}
3.74/1.86	
3.74/1.86	  5: evalndecrreturnin->evalndecrstop, Arg_0: 1 {O(1)}
3.74/1.86	
3.74/1.86	  0: evalndecrstart->evalndecrentryin, Arg_0: Arg_0 {O(n)}
3.74/1.86	
3.74/1.86	
3.74/1.86	----------------------------------------
3.74/1.86	
3.74/1.86	(2)
3.74/1.86	BOUNDS(1, max(3, 2 + Arg_0) + max(1, 1 + Arg_0))
3.74/1.86	
3.74/1.86	----------------------------------------
3.74/1.86	
3.74/1.86	(3) Loat Proof (FINISHED)
3.74/1.86	
3.74/1.86	
3.74/1.86	### Pre-processing the ITS problem ###
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Initial linear ITS problem
3.74/1.86	
3.74/1.86	   Start location: evalndecrstart
3.74/1.86	
3.74/1.86	      0: evalndecrstart -> evalndecrentryin : [], cost: 1
3.74/1.86	
3.74/1.86	      1: evalndecrentryin -> evalndecrbb1in : A'=-1+A, [], cost: 1
3.74/1.86	
3.74/1.86	      2: evalndecrbb1in -> evalndecrbbin : [ A>=2 ], cost: 1
3.74/1.86	
3.74/1.86	      3: evalndecrbb1in -> evalndecrreturnin : [ 1>=A ], cost: 1
3.74/1.86	
3.74/1.86	      4: evalndecrbbin -> evalndecrbb1in : A'=-1+A, [], cost: 1
3.74/1.86	
3.74/1.86	      5: evalndecrreturnin -> evalndecrstop : [], cost: 1
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Removed unreachable and leaf rules:
3.74/1.86	
3.74/1.86	   Start location: evalndecrstart
3.74/1.86	
3.74/1.86	      0: evalndecrstart -> evalndecrentryin : [], cost: 1
3.74/1.86	
3.74/1.86	      1: evalndecrentryin -> evalndecrbb1in : A'=-1+A, [], cost: 1
3.74/1.86	
3.74/1.86	      2: evalndecrbb1in -> evalndecrbbin : [ A>=2 ], cost: 1
3.74/1.86	
3.74/1.86	      4: evalndecrbbin -> evalndecrbb1in : A'=-1+A, [], cost: 1
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	### Simplification by acceleration and chaining ###
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Eliminated locations (on linear paths):
3.74/1.86	
3.74/1.86	   Start location: evalndecrstart
3.74/1.86	
3.74/1.86	      6: evalndecrstart -> evalndecrbb1in : A'=-1+A, [], cost: 2
3.74/1.86	
3.74/1.86	      7: evalndecrbb1in -> evalndecrbb1in : A'=-1+A, [ A>=2 ], cost: 2
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Accelerating simple loops of location 2.
3.74/1.86	
3.74/1.86	   Accelerating the following rules:
3.74/1.86	
3.74/1.86	      7: evalndecrbb1in -> evalndecrbb1in : A'=-1+A, [ A>=2 ], cost: 2
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	   Accelerated rule 7 with metering function -1+A, yielding the new rule 8.
3.74/1.86	
3.74/1.86	   Removing the simple loops: 7.
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Accelerated all simple loops using metering functions (where possible):
3.74/1.86	
3.74/1.86	   Start location: evalndecrstart
3.74/1.86	
3.74/1.86	      6: evalndecrstart -> evalndecrbb1in : A'=-1+A, [], cost: 2
3.74/1.86	
3.74/1.86	      8: evalndecrbb1in -> evalndecrbb1in : A'=1, [ A>=2 ], cost: -2+2*A
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Chained accelerated rules (with incoming rules):
3.74/1.86	
3.74/1.86	   Start location: evalndecrstart
3.74/1.86	
3.74/1.86	      6: evalndecrstart -> evalndecrbb1in : A'=-1+A, [], cost: 2
3.74/1.86	
3.74/1.86	      9: evalndecrstart -> evalndecrbb1in : A'=1, [ -1+A>=2 ], cost: -2+2*A
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Removed unreachable locations (and leaf rules with constant cost):
3.74/1.86	
3.74/1.86	   Start location: evalndecrstart
3.74/1.86	
3.74/1.86	      9: evalndecrstart -> evalndecrbb1in : A'=1, [ -1+A>=2 ], cost: -2+2*A
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	### Computing asymptotic complexity ###
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Fully simplified ITS problem
3.74/1.86	
3.74/1.86	   Start location: evalndecrstart
3.74/1.86	
3.74/1.86	      9: evalndecrstart -> evalndecrbb1in : A'=1, [ -1+A>=2 ], cost: -2+2*A
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Computing asymptotic complexity for rule 9
3.74/1.86	
3.74/1.86	   Solved the limit problem by the following transformations:
3.74/1.86	
3.74/1.86	      Created initial limit problem:
3.74/1.86	
3.74/1.86	      -2+2*A (+), -2+A (+/+!) [not solved]
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	      removing all constraints (solved by SMT)
3.74/1.86	
3.74/1.86	      resulting limit problem: [solved]
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	      applying transformation rule (C) using substitution {A==n}
3.74/1.86	
3.74/1.86	      resulting limit problem:
3.74/1.86	
3.74/1.86	      [solved]
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	   Solution:
3.74/1.86	
3.74/1.86	      A / n
3.74/1.86	
3.74/1.86	   Resulting cost -2+2*n has complexity: Poly(n^1)
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Found new complexity Poly(n^1).
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	Obtained the following overall complexity (w.r.t. the length of the input n):
3.74/1.86	
3.74/1.86	   Complexity:  Poly(n^1)
3.74/1.86	
3.74/1.86	   Cpx degree:  1
3.74/1.86	
3.74/1.86	   Solved cost: -2+2*n
3.74/1.86	
3.74/1.86	   Rule cost:   -2+2*A
3.74/1.86	
3.74/1.86	   Rule guard:  [ -1+A>=2 ]
3.74/1.86	
3.74/1.86	
3.74/1.86	
3.74/1.86	WORST_CASE(Omega(n^1),?)
3.74/1.86	
3.74/1.86	
3.74/1.86	----------------------------------------
3.74/1.86	
3.74/1.86	(4)
3.74/1.86	BOUNDS(n^1, INF)
3.74/1.89	EOF