3.87/1.85	WORST_CASE(Omega(n^1), O(n^1))
3.95/1.86	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.95/1.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.95/1.86	
3.95/1.86	
3.95/1.86	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
3.95/1.86	
3.95/1.86	(0) CpxIntTrs
3.95/1.86	(1) Koat Proof [FINISHED, 118 ms]
3.95/1.86	(2) BOUNDS(1, n^1)
3.95/1.86	(3) Loat Proof [FINISHED, 229 ms]
3.95/1.86	(4) BOUNDS(n^1, INF)
3.95/1.86	
3.95/1.86	
3.95/1.86	----------------------------------------
3.95/1.86	
3.95/1.86	(0)
3.95/1.86	Obligation:
3.95/1.86	Complexity Int TRS consisting of the following rules:
3.95/1.86	start(A, B, C, D) -> Com_1(stop(A, B, C, D)) :|: 1 >= A && B >= C && B <= C && D >= A && D <= A
3.95/1.86	start(A, B, C, D) -> Com_1(lbl32(A, B, C, D - 1)) :|: A >= 2 && B >= C && B <= C && D >= A && D <= A
3.95/1.86	lbl32(A, B, C, D) -> Com_1(stop(A, B, C, D)) :|: A >= 2 && D >= 1 && D <= 1 && B >= C && B <= C
3.95/1.86	lbl32(A, B, C, D) -> Com_1(lbl32(A, B, C, D - 1)) :|: D >= 2 && D >= 1 && A >= D + 1 && B >= C && B <= C
3.95/1.86	start0(A, B, C, D) -> Com_1(start(A, C, C, A)) :|: TRUE
3.95/1.86	
3.95/1.86	The start-symbols are:[start0_4]
3.95/1.86	
3.95/1.86	
3.95/1.86	----------------------------------------
3.95/1.86	
3.95/1.86	(1) Koat Proof (FINISHED)
3.95/1.86	YES(?, ar_0 + 4)
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Initial complexity problem:
3.95/1.86	
3.95/1.86	1:	T:
3.95/1.86	
3.95/1.86			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 1 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
3.95/1.86	
3.95/1.86			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl32(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_0 >= 2 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
3.95/1.86	
3.95/1.86			(Comp: ?, Cost: 1)    lbl32(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 2 /\ ar_3 = 1 /\ ar_1 = ar_2 ]
3.95/1.86	
3.95/1.86			(Comp: ?, Cost: 1)    lbl32(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl32(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 2 /\ ar_3 >= 1 /\ ar_0 >= ar_3 + 1 /\ ar_1 = ar_2 ]
3.95/1.86	
3.95/1.86			(Comp: ?, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3) -> Com_1(start(ar_0, ar_2, ar_2, ar_0))
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.95/1.86	
3.95/1.86		start location:	koat_start
3.95/1.86	
3.95/1.86		leaf cost:	0
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.95/1.86	
3.95/1.86	2:	T:
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 1 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl32(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_0 >= 2 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
3.95/1.86	
3.95/1.86			(Comp: ?, Cost: 1)    lbl32(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 2 /\ ar_3 = 1 /\ ar_1 = ar_2 ]
3.95/1.86	
3.95/1.86			(Comp: ?, Cost: 1)    lbl32(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl32(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 2 /\ ar_3 >= 1 /\ ar_0 >= ar_3 + 1 /\ ar_1 = ar_2 ]
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3) -> Com_1(start(ar_0, ar_2, ar_2, ar_0))
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.95/1.86	
3.95/1.86		start location:	koat_start
3.95/1.86	
3.95/1.86		leaf cost:	0
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	A polynomial rank function with
3.95/1.86	
3.95/1.86		Pol(start) = 1
3.95/1.86	
3.95/1.86		Pol(stop) = 0
3.95/1.86	
3.95/1.86		Pol(lbl32) = 1
3.95/1.86	
3.95/1.86		Pol(start0) = 1
3.95/1.86	
3.95/1.86		Pol(koat_start) = 1
3.95/1.86	
3.95/1.86	orients all transitions weakly and the transition
3.95/1.86	
3.95/1.86		lbl32(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 2 /\ ar_3 = 1 /\ ar_1 = ar_2 ]
3.95/1.86	
3.95/1.86	strictly and produces the following problem:
3.95/1.86	
3.95/1.86	3:	T:
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 1 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl32(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_0 >= 2 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 1)    lbl32(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 2 /\ ar_3 = 1 /\ ar_1 = ar_2 ]
3.95/1.86	
3.95/1.86			(Comp: ?, Cost: 1)    lbl32(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl32(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 2 /\ ar_3 >= 1 /\ ar_0 >= ar_3 + 1 /\ ar_1 = ar_2 ]
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3) -> Com_1(start(ar_0, ar_2, ar_2, ar_0))
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.95/1.86	
3.95/1.86		start location:	koat_start
3.95/1.86	
3.95/1.86		leaf cost:	0
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	A polynomial rank function with
3.95/1.86	
3.95/1.86		Pol(start) = V_1
3.95/1.86	
3.95/1.86		Pol(stop) = V_4
3.95/1.86	
3.95/1.86		Pol(lbl32) = V_4
3.95/1.86	
3.95/1.86		Pol(start0) = V_1
3.95/1.86	
3.95/1.86		Pol(koat_start) = V_1
3.95/1.86	
3.95/1.86	orients all transitions weakly and the transition
3.95/1.86	
3.95/1.86		lbl32(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl32(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 2 /\ ar_3 >= 1 /\ ar_0 >= ar_3 + 1 /\ ar_1 = ar_2 ]
3.95/1.86	
3.95/1.86	strictly and produces the following problem:
3.95/1.86	
3.95/1.86	4:	T:
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 1)       start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 1 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 1)       start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl32(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_0 >= 2 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 1)       lbl32(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 2 /\ ar_3 = 1 /\ ar_1 = ar_2 ]
3.95/1.86	
3.95/1.86			(Comp: ar_0, Cost: 1)    lbl32(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl32(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 2 /\ ar_3 >= 1 /\ ar_0 >= ar_3 + 1 /\ ar_1 = ar_2 ]
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 1)       start0(ar_0, ar_1, ar_2, ar_3) -> Com_1(start(ar_0, ar_2, ar_2, ar_0))
3.95/1.86	
3.95/1.86			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.95/1.86	
3.95/1.86		start location:	koat_start
3.95/1.86	
3.95/1.86		leaf cost:	0
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Complexity upper bound ar_0 + 4
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Time: 0.101 sec (SMT: 0.092 sec)
3.95/1.86	
3.95/1.86	
3.95/1.86	----------------------------------------
3.95/1.86	
3.95/1.86	(2)
3.95/1.86	BOUNDS(1, n^1)
3.95/1.86	
3.95/1.86	----------------------------------------
3.95/1.86	
3.95/1.86	(3) Loat Proof (FINISHED)
3.95/1.86	
3.95/1.86	
3.95/1.86	### Pre-processing the ITS problem ###
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Initial linear ITS problem
3.95/1.86	
3.95/1.86	   Start location: start0
3.95/1.86	
3.95/1.86	      0: start -> stop : [ 1>=A && B==C && D==A ], cost: 1
3.95/1.86	
3.95/1.86	      1: start -> lbl32 : D'=-1+D, [ A>=2 && B==C && D==A ], cost: 1
3.95/1.86	
3.95/1.86	      2: lbl32 -> stop : [ A>=2 && D==1 && B==C ], cost: 1
3.95/1.86	
3.95/1.86	      3: lbl32 -> lbl32 : D'=-1+D, [ D>=2 && D>=1 && A>=1+D && B==C ], cost: 1
3.95/1.86	
3.95/1.86	      4: start0 -> start : B'=C, D'=A, [], cost: 1
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Removed unreachable and leaf rules:
3.95/1.86	
3.95/1.86	   Start location: start0
3.95/1.86	
3.95/1.86	      1: start -> lbl32 : D'=-1+D, [ A>=2 && B==C && D==A ], cost: 1
3.95/1.86	
3.95/1.86	      3: lbl32 -> lbl32 : D'=-1+D, [ D>=2 && D>=1 && A>=1+D && B==C ], cost: 1
3.95/1.86	
3.95/1.86	      4: start0 -> start : B'=C, D'=A, [], cost: 1
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Simplified all rules, resulting in:
3.95/1.86	
3.95/1.86	   Start location: start0
3.95/1.86	
3.95/1.86	      1: start -> lbl32 : D'=-1+D, [ A>=2 && B==C && D==A ], cost: 1
3.95/1.86	
3.95/1.86	      3: lbl32 -> lbl32 : D'=-1+D, [ D>=2 && A>=1+D && B==C ], cost: 1
3.95/1.86	
3.95/1.86	      4: start0 -> start : B'=C, D'=A, [], cost: 1
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	### Simplification by acceleration and chaining ###
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Accelerating simple loops of location 1.
3.95/1.86	
3.95/1.86	   Accelerating the following rules:
3.95/1.86	
3.95/1.86	      3: lbl32 -> lbl32 : D'=-1+D, [ D>=2 && A>=1+D && B==C ], cost: 1
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	   Accelerated rule 3 with metering function -1+D, yielding the new rule 5.
3.95/1.86	
3.95/1.86	   Removing the simple loops: 3.
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Accelerated all simple loops using metering functions (where possible):
3.95/1.86	
3.95/1.86	   Start location: start0
3.95/1.86	
3.95/1.86	      1: start -> lbl32 : D'=-1+D, [ A>=2 && B==C && D==A ], cost: 1
3.95/1.86	
3.95/1.86	      5: lbl32 -> lbl32 : D'=1, [ D>=2 && A>=1+D && B==C ], cost: -1+D
3.95/1.86	
3.95/1.86	      4: start0 -> start : B'=C, D'=A, [], cost: 1
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Chained accelerated rules (with incoming rules):
3.95/1.86	
3.95/1.86	   Start location: start0
3.95/1.86	
3.95/1.86	      1: start -> lbl32 : D'=-1+D, [ A>=2 && B==C && D==A ], cost: 1
3.95/1.86	
3.95/1.86	      6: start -> lbl32 : D'=1, [ A>=2 && B==C && D==A && -1+D>=2 ], cost: -1+D
3.95/1.86	
3.95/1.86	      4: start0 -> start : B'=C, D'=A, [], cost: 1
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Removed unreachable locations (and leaf rules with constant cost):
3.95/1.86	
3.95/1.86	   Start location: start0
3.95/1.86	
3.95/1.86	      6: start -> lbl32 : D'=1, [ A>=2 && B==C && D==A && -1+D>=2 ], cost: -1+D
3.95/1.86	
3.95/1.86	      4: start0 -> start : B'=C, D'=A, [], cost: 1
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Eliminated locations (on linear paths):
3.95/1.86	
3.95/1.86	   Start location: start0
3.95/1.86	
3.95/1.86	      7: start0 -> lbl32 : B'=C, D'=1, [ -1+A>=2 ], cost: A
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	### Computing asymptotic complexity ###
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Fully simplified ITS problem
3.95/1.86	
3.95/1.86	   Start location: start0
3.95/1.86	
3.95/1.86	      7: start0 -> lbl32 : B'=C, D'=1, [ -1+A>=2 ], cost: A
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Computing asymptotic complexity for rule 7
3.95/1.86	
3.95/1.86	   Solved the limit problem by the following transformations:
3.95/1.86	
3.95/1.86	      Created initial limit problem:
3.95/1.86	
3.95/1.86	      A (+), -2+A (+/+!) [not solved]
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	      removing all constraints (solved by SMT)
3.95/1.86	
3.95/1.86	      resulting limit problem: [solved]
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	      applying transformation rule (C) using substitution {A==n}
3.95/1.86	
3.95/1.86	      resulting limit problem:
3.95/1.86	
3.95/1.86	      [solved]
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	   Solution:
3.95/1.86	
3.95/1.86	      A / n
3.95/1.86	
3.95/1.86	   Resulting cost n has complexity: Poly(n^1)
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Found new complexity Poly(n^1).
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	Obtained the following overall complexity (w.r.t. the length of the input n):
3.95/1.86	
3.95/1.86	   Complexity:  Poly(n^1)
3.95/1.86	
3.95/1.86	   Cpx degree:  1
3.95/1.86	
3.95/1.86	   Solved cost: n
3.95/1.86	
3.95/1.86	   Rule cost:   A
3.95/1.86	
3.95/1.86	   Rule guard:  [ -1+A>=2 ]
3.95/1.86	
3.95/1.86	
3.95/1.86	
3.95/1.86	WORST_CASE(Omega(n^1),?)
3.95/1.86	
3.95/1.86	
3.95/1.86	----------------------------------------
3.95/1.86	
3.95/1.86	(4)
3.95/1.86	BOUNDS(n^1, INF)
3.97/2.22	EOF