3.51/1.84	WORST_CASE(Omega(n^1), O(n^1))
3.51/1.85	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.51/1.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.51/1.85	
3.51/1.85	
3.51/1.85	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
3.51/1.85	
3.51/1.85	(0) CpxIntTrs
3.51/1.85	(1) Koat Proof [FINISHED, 6 ms]
3.51/1.85	(2) BOUNDS(1, n^1)
3.51/1.85	(3) Loat Proof [FINISHED, 222 ms]
3.51/1.85	(4) BOUNDS(n^1, INF)
3.51/1.85	
3.51/1.85	
3.51/1.85	----------------------------------------
3.51/1.85	
3.51/1.85	(0)
3.51/1.85	Obligation:
3.51/1.85	Complexity Int TRS consisting of the following rules:
3.51/1.85	evalfstart(A, B) -> Com_1(evalfentryin(A, B)) :|: TRUE
3.51/1.85	evalfentryin(A, B) -> Com_1(evalfbb1in(B, A)) :|: TRUE
3.51/1.85	evalfbb1in(A, B) -> Com_1(evalfbbin(A, B)) :|: A >= B
3.51/1.85	evalfbb1in(A, B) -> Com_1(evalfreturnin(A, B)) :|: B >= A + 1
3.51/1.85	evalfbbin(A, B) -> Com_1(evalfbb1in(A, B + 1)) :|: TRUE
3.51/1.85	evalfreturnin(A, B) -> Com_1(evalfstop(A, B)) :|: TRUE
3.51/1.85	
3.51/1.85	The start-symbols are:[evalfstart_2]
3.51/1.85	
3.51/1.85	
3.51/1.85	----------------------------------------
3.51/1.85	
3.51/1.85	(1) Koat Proof (FINISHED)
3.51/1.85	YES(?, 2*ar_0 + 2*ar_1 + 8)
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Initial complexity problem:
3.51/1.85	
3.51/1.85	1:	T:
3.51/1.85	
3.51/1.85			(Comp: ?, Cost: 1)    evalfstart(ar_0, ar_1) -> Com_1(evalfentryin(ar_0, ar_1))
3.51/1.85	
3.51/1.85			(Comp: ?, Cost: 1)    evalfentryin(ar_0, ar_1) -> Com_1(evalfbb1in(ar_1, ar_0))
3.51/1.85	
3.51/1.85			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1) -> Com_1(evalfbbin(ar_0, ar_1)) [ ar_0 >= ar_1 ]
3.51/1.85	
3.51/1.85			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1) -> Com_1(evalfreturnin(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
3.51/1.85	
3.51/1.85			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1) -> Com_1(evalfbb1in(ar_0, ar_1 + 1))
3.51/1.85	
3.51/1.85			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1) -> Com_1(evalfstop(ar_0, ar_1))
3.51/1.85	
3.51/1.85			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfstart(ar_0, ar_1)) [ 0 <= 0 ]
3.51/1.85	
3.51/1.85		start location:	koat_start
3.51/1.85	
3.51/1.85		leaf cost:	0
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.51/1.85	
3.51/1.85	2:	T:
3.51/1.85	
3.51/1.85			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1) -> Com_1(evalfentryin(ar_0, ar_1))
3.51/1.85	
3.51/1.85			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1) -> Com_1(evalfbb1in(ar_1, ar_0))
3.51/1.85	
3.51/1.85			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1) -> Com_1(evalfbbin(ar_0, ar_1)) [ ar_0 >= ar_1 ]
3.51/1.85	
3.51/1.85			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1) -> Com_1(evalfreturnin(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
3.51/1.85	
3.51/1.85			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1) -> Com_1(evalfbb1in(ar_0, ar_1 + 1))
3.51/1.85	
3.51/1.85			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1) -> Com_1(evalfstop(ar_0, ar_1))
3.51/1.85	
3.51/1.85			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfstart(ar_0, ar_1)) [ 0 <= 0 ]
3.51/1.85	
3.51/1.85		start location:	koat_start
3.51/1.85	
3.51/1.85		leaf cost:	0
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	A polynomial rank function with
3.51/1.85	
3.51/1.85		Pol(evalfstart) = 2
3.51/1.85	
3.51/1.85		Pol(evalfentryin) = 2
3.51/1.85	
3.51/1.85		Pol(evalfbb1in) = 2
3.51/1.85	
3.51/1.85		Pol(evalfbbin) = 2
3.51/1.85	
3.51/1.85		Pol(evalfreturnin) = 1
3.51/1.85	
3.51/1.85		Pol(evalfstop) = 0
3.51/1.85	
3.51/1.85		Pol(koat_start) = 2
3.51/1.85	
3.51/1.85	orients all transitions weakly and the transitions
3.51/1.85	
3.51/1.85		evalfreturnin(ar_0, ar_1) -> Com_1(evalfstop(ar_0, ar_1))
3.51/1.85	
3.51/1.85		evalfbb1in(ar_0, ar_1) -> Com_1(evalfreturnin(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
3.51/1.85	
3.51/1.85	strictly and produces the following problem:
3.51/1.85	
3.51/1.85	3:	T:
3.51/1.85	
3.51/1.85			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1) -> Com_1(evalfentryin(ar_0, ar_1))
3.51/1.85	
3.51/1.85			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1) -> Com_1(evalfbb1in(ar_1, ar_0))
3.51/1.85	
3.51/1.85			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1) -> Com_1(evalfbbin(ar_0, ar_1)) [ ar_0 >= ar_1 ]
3.51/1.85	
3.51/1.85			(Comp: 2, Cost: 1)    evalfbb1in(ar_0, ar_1) -> Com_1(evalfreturnin(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
3.51/1.85	
3.51/1.85			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1) -> Com_1(evalfbb1in(ar_0, ar_1 + 1))
3.51/1.85	
3.51/1.85			(Comp: 2, Cost: 1)    evalfreturnin(ar_0, ar_1) -> Com_1(evalfstop(ar_0, ar_1))
3.51/1.85	
3.51/1.85			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfstart(ar_0, ar_1)) [ 0 <= 0 ]
3.51/1.85	
3.51/1.85		start location:	koat_start
3.51/1.85	
3.51/1.85		leaf cost:	0
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	A polynomial rank function with
3.51/1.85	
3.51/1.85		Pol(evalfstart) = -V_1 + V_2 + 1
3.51/1.85	
3.51/1.85		Pol(evalfentryin) = -V_1 + V_2 + 1
3.51/1.85	
3.51/1.85		Pol(evalfbb1in) = V_1 - V_2 + 1
3.51/1.85	
3.51/1.85		Pol(evalfbbin) = V_1 - V_2
3.51/1.85	
3.51/1.85		Pol(evalfreturnin) = V_1 - V_2
3.51/1.85	
3.51/1.85		Pol(evalfstop) = V_1 - V_2
3.51/1.85	
3.51/1.85		Pol(koat_start) = -V_1 + V_2 + 1
3.51/1.85	
3.51/1.85	orients all transitions weakly and the transition
3.51/1.85	
3.51/1.85		evalfbb1in(ar_0, ar_1) -> Com_1(evalfbbin(ar_0, ar_1)) [ ar_0 >= ar_1 ]
3.51/1.85	
3.51/1.85	strictly and produces the following problem:
3.51/1.85	
3.51/1.85	4:	T:
3.51/1.85	
3.51/1.85			(Comp: 1, Cost: 1)                  evalfstart(ar_0, ar_1) -> Com_1(evalfentryin(ar_0, ar_1))
3.51/1.85	
3.51/1.85			(Comp: 1, Cost: 1)                  evalfentryin(ar_0, ar_1) -> Com_1(evalfbb1in(ar_1, ar_0))
3.51/1.85	
3.51/1.85			(Comp: ar_0 + ar_1 + 1, Cost: 1)    evalfbb1in(ar_0, ar_1) -> Com_1(evalfbbin(ar_0, ar_1)) [ ar_0 >= ar_1 ]
3.51/1.85	
3.51/1.85			(Comp: 2, Cost: 1)                  evalfbb1in(ar_0, ar_1) -> Com_1(evalfreturnin(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
3.51/1.85	
3.51/1.85			(Comp: ?, Cost: 1)                  evalfbbin(ar_0, ar_1) -> Com_1(evalfbb1in(ar_0, ar_1 + 1))
3.51/1.85	
3.51/1.85			(Comp: 2, Cost: 1)                  evalfreturnin(ar_0, ar_1) -> Com_1(evalfstop(ar_0, ar_1))
3.51/1.85	
3.51/1.85			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1) -> Com_1(evalfstart(ar_0, ar_1)) [ 0 <= 0 ]
3.51/1.85	
3.51/1.85		start location:	koat_start
3.51/1.85	
3.51/1.85		leaf cost:	0
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Repeatedly propagating knowledge in problem 4 produces the following problem:
3.51/1.85	
3.51/1.85	5:	T:
3.51/1.85	
3.51/1.85			(Comp: 1, Cost: 1)                  evalfstart(ar_0, ar_1) -> Com_1(evalfentryin(ar_0, ar_1))
3.51/1.85	
3.51/1.85			(Comp: 1, Cost: 1)                  evalfentryin(ar_0, ar_1) -> Com_1(evalfbb1in(ar_1, ar_0))
3.51/1.85	
3.51/1.85			(Comp: ar_0 + ar_1 + 1, Cost: 1)    evalfbb1in(ar_0, ar_1) -> Com_1(evalfbbin(ar_0, ar_1)) [ ar_0 >= ar_1 ]
3.51/1.85	
3.51/1.85			(Comp: 2, Cost: 1)                  evalfbb1in(ar_0, ar_1) -> Com_1(evalfreturnin(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
3.51/1.85	
3.51/1.85			(Comp: ar_0 + ar_1 + 1, Cost: 1)    evalfbbin(ar_0, ar_1) -> Com_1(evalfbb1in(ar_0, ar_1 + 1))
3.51/1.85	
3.51/1.85			(Comp: 2, Cost: 1)                  evalfreturnin(ar_0, ar_1) -> Com_1(evalfstop(ar_0, ar_1))
3.51/1.85	
3.51/1.85			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1) -> Com_1(evalfstart(ar_0, ar_1)) [ 0 <= 0 ]
3.51/1.85	
3.51/1.85		start location:	koat_start
3.51/1.85	
3.51/1.85		leaf cost:	0
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Complexity upper bound 2*ar_0 + 2*ar_1 + 8
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Time: 0.055 sec (SMT: 0.052 sec)
3.51/1.85	
3.51/1.85	
3.51/1.85	----------------------------------------
3.51/1.85	
3.51/1.85	(2)
3.51/1.85	BOUNDS(1, n^1)
3.51/1.85	
3.51/1.85	----------------------------------------
3.51/1.85	
3.51/1.85	(3) Loat Proof (FINISHED)
3.51/1.85	
3.51/1.85	
3.51/1.85	### Pre-processing the ITS problem ###
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Initial linear ITS problem
3.51/1.85	
3.51/1.85	   Start location: evalfstart
3.51/1.85	
3.51/1.85	      0: evalfstart -> evalfentryin : [], cost: 1
3.51/1.85	
3.51/1.85	      1: evalfentryin -> evalfbb1in : A'=B, B'=A, [], cost: 1
3.51/1.85	
3.51/1.85	      2: evalfbb1in -> evalfbbin : [ A>=B ], cost: 1
3.51/1.85	
3.51/1.85	      3: evalfbb1in -> evalfreturnin : [ B>=1+A ], cost: 1
3.51/1.85	
3.51/1.85	      4: evalfbbin -> evalfbb1in : B'=1+B, [], cost: 1
3.51/1.85	
3.51/1.85	      5: evalfreturnin -> evalfstop : [], cost: 1
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Removed unreachable and leaf rules:
3.51/1.85	
3.51/1.85	   Start location: evalfstart
3.51/1.85	
3.51/1.85	      0: evalfstart -> evalfentryin : [], cost: 1
3.51/1.85	
3.51/1.85	      1: evalfentryin -> evalfbb1in : A'=B, B'=A, [], cost: 1
3.51/1.85	
3.51/1.85	      2: evalfbb1in -> evalfbbin : [ A>=B ], cost: 1
3.51/1.85	
3.51/1.85	      4: evalfbbin -> evalfbb1in : B'=1+B, [], cost: 1
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	### Simplification by acceleration and chaining ###
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Eliminated locations (on linear paths):
3.51/1.85	
3.51/1.85	   Start location: evalfstart
3.51/1.85	
3.51/1.85	      6: evalfstart -> evalfbb1in : A'=B, B'=A, [], cost: 2
3.51/1.85	
3.51/1.85	      7: evalfbb1in -> evalfbb1in : B'=1+B, [ A>=B ], cost: 2
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Accelerating simple loops of location 2.
3.51/1.85	
3.51/1.85	   Accelerating the following rules:
3.51/1.85	
3.51/1.85	      7: evalfbb1in -> evalfbb1in : B'=1+B, [ A>=B ], cost: 2
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	   Accelerated rule 7 with metering function 1+A-B, yielding the new rule 8.
3.51/1.85	
3.51/1.85	   Removing the simple loops: 7.
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Accelerated all simple loops using metering functions (where possible):
3.51/1.85	
3.51/1.85	   Start location: evalfstart
3.51/1.85	
3.51/1.85	      6: evalfstart -> evalfbb1in : A'=B, B'=A, [], cost: 2
3.51/1.85	
3.51/1.85	      8: evalfbb1in -> evalfbb1in : B'=1+A, [ A>=B ], cost: 2+2*A-2*B
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Chained accelerated rules (with incoming rules):
3.51/1.85	
3.51/1.85	   Start location: evalfstart
3.51/1.85	
3.51/1.85	      6: evalfstart -> evalfbb1in : A'=B, B'=A, [], cost: 2
3.51/1.85	
3.51/1.85	      9: evalfstart -> evalfbb1in : A'=B, B'=1+B, [ B>=A ], cost: 4-2*A+2*B
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Removed unreachable locations (and leaf rules with constant cost):
3.51/1.85	
3.51/1.85	   Start location: evalfstart
3.51/1.85	
3.51/1.85	      9: evalfstart -> evalfbb1in : A'=B, B'=1+B, [ B>=A ], cost: 4-2*A+2*B
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	### Computing asymptotic complexity ###
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Fully simplified ITS problem
3.51/1.85	
3.51/1.85	   Start location: evalfstart
3.51/1.85	
3.51/1.85	      9: evalfstart -> evalfbb1in : A'=B, B'=1+B, [ B>=A ], cost: 4-2*A+2*B
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Computing asymptotic complexity for rule 9
3.51/1.85	
3.51/1.85	   Solved the limit problem by the following transformations:
3.51/1.85	
3.51/1.85	      Created initial limit problem:
3.51/1.85	
3.51/1.85	      4-2*A+2*B (+), 1-A+B (+/+!) [not solved]
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	      removing all constraints (solved by SMT)
3.51/1.85	
3.51/1.85	      resulting limit problem: [solved]
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	      applying transformation rule (C) using substitution {A==0,B==n}
3.51/1.85	
3.51/1.85	      resulting limit problem:
3.51/1.85	
3.51/1.85	      [solved]
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	   Solution:
3.51/1.85	
3.51/1.85	      A / 0
3.51/1.85	
3.51/1.85	      B / n
3.51/1.85	
3.51/1.85	   Resulting cost 4+2*n has complexity: Poly(n^1)
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Found new complexity Poly(n^1).
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	Obtained the following overall complexity (w.r.t. the length of the input n):
3.51/1.85	
3.51/1.85	   Complexity:  Poly(n^1)
3.51/1.85	
3.51/1.85	   Cpx degree:  1
3.51/1.85	
3.51/1.85	   Solved cost: 4+2*n
3.51/1.85	
3.51/1.85	   Rule cost:   4-2*A+2*B
3.51/1.85	
3.51/1.85	   Rule guard:  [ B>=A ]
3.51/1.85	
3.51/1.85	
3.51/1.85	
3.51/1.85	WORST_CASE(Omega(n^1),?)
3.51/1.85	
3.51/1.85	
3.51/1.85	----------------------------------------
3.51/1.85	
3.51/1.85	(4)
3.51/1.85	BOUNDS(n^1, INF)
4.04/1.88	EOF