4.07/2.05	WORST_CASE(Omega(n^1), O(n^1))
4.17/2.21	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.17/2.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.17/2.21	
4.17/2.21	
4.17/2.21	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
4.17/2.21	
4.17/2.21	(0) CpxIntTrs
4.17/2.21	(1) Koat Proof [FINISHED, 30 ms]
4.17/2.21	(2) BOUNDS(1, n^1)
4.17/2.21	(3) Loat Proof [FINISHED, 434 ms]
4.17/2.21	(4) BOUNDS(n^1, INF)
4.17/2.21	
4.17/2.21	
4.17/2.21	----------------------------------------
4.17/2.21	
4.17/2.21	(0)
4.17/2.21	Obligation:
4.17/2.21	Complexity Int TRS consisting of the following rules:
4.17/2.21	eval1(A, B, C) -> Com_1(eval2(A, B, C)) :|: A >= B + 1
4.17/2.21	eval2(A, B, C) -> Com_1(eval2(A, B, C - 1)) :|: A >= B + 1 && C >= B + 1
4.17/2.21	eval2(A, B, C) -> Com_1(eval1(A - 1, B, C)) :|: A >= B + 1 && B >= C
4.17/2.21	start(A, B, C) -> Com_1(eval1(A, B, C)) :|: TRUE
4.17/2.21	
4.17/2.21	The start-symbols are:[start_3]
4.17/2.21	
4.17/2.21	
4.17/2.21	----------------------------------------
4.17/2.21	
4.17/2.21	(1) Koat Proof (FINISHED)
4.17/2.21	YES(?, 4*ar_0 + 5*ar_1 + ar_2 + 1)
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Initial complexity problem:
4.17/2.21	
4.17/2.21	1:	T:
4.17/2.21	
4.17/2.21			(Comp: ?, Cost: 1)    eval1(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
4.17/2.21	
4.17/2.21			(Comp: ?, Cost: 1)    eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 /\ ar_2 >= ar_1 + 1 ]
4.17/2.21	
4.17/2.21			(Comp: ?, Cost: 1)    eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0 - 1, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 /\ ar_1 >= ar_2 ]
4.17/2.21	
4.17/2.21			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2))
4.17/2.21	
4.17/2.21			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.17/2.21	
4.17/2.21		start location:	koat_start
4.17/2.21	
4.17/2.21		leaf cost:	0
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Repeatedly propagating knowledge in problem 1 produces the following problem:
4.17/2.21	
4.17/2.21	2:	T:
4.17/2.21	
4.17/2.21			(Comp: ?, Cost: 1)    eval1(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
4.17/2.21	
4.17/2.21			(Comp: ?, Cost: 1)    eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 /\ ar_2 >= ar_1 + 1 ]
4.17/2.21	
4.17/2.21			(Comp: ?, Cost: 1)    eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0 - 1, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 /\ ar_1 >= ar_2 ]
4.17/2.21	
4.17/2.21			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2))
4.17/2.21	
4.17/2.21			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.17/2.21	
4.17/2.21		start location:	koat_start
4.17/2.21	
4.17/2.21		leaf cost:	0
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	A polynomial rank function with
4.17/2.21	
4.17/2.21		Pol(eval1) = 2*V_1 - 2*V_2
4.17/2.21	
4.17/2.21		Pol(eval2) = 2*V_1 - 2*V_2 - 1
4.17/2.21	
4.17/2.21		Pol(start) = 2*V_1 - 2*V_2
4.17/2.21	
4.17/2.21		Pol(koat_start) = 2*V_1 - 2*V_2
4.17/2.21	
4.17/2.21	orients all transitions weakly and the transitions
4.17/2.21	
4.17/2.21		eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0 - 1, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 /\ ar_1 >= ar_2 ]
4.17/2.21	
4.17/2.21		eval1(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
4.17/2.21	
4.17/2.21	strictly and produces the following problem:
4.17/2.21	
4.17/2.21	3:	T:
4.17/2.21	
4.17/2.21			(Comp: 2*ar_0 + 2*ar_1, Cost: 1)    eval1(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
4.17/2.21	
4.17/2.21			(Comp: ?, Cost: 1)                  eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 /\ ar_2 >= ar_1 + 1 ]
4.17/2.21	
4.17/2.21			(Comp: 2*ar_0 + 2*ar_1, Cost: 1)    eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0 - 1, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 /\ ar_1 >= ar_2 ]
4.17/2.21	
4.17/2.21			(Comp: 1, Cost: 1)                  start(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2))
4.17/2.21	
4.17/2.21			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2) -> Com_1(start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.17/2.21	
4.17/2.21		start location:	koat_start
4.17/2.21	
4.17/2.21		leaf cost:	0
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	A polynomial rank function with
4.17/2.21	
4.17/2.21		Pol(eval1) = -V_2 + V_3
4.17/2.21	
4.17/2.21		Pol(eval2) = -V_2 + V_3
4.17/2.21	
4.17/2.21		Pol(start) = -V_2 + V_3
4.17/2.21	
4.17/2.21		Pol(koat_start) = -V_2 + V_3
4.17/2.21	
4.17/2.21	orients all transitions weakly and the transition
4.17/2.21	
4.17/2.21		eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 /\ ar_2 >= ar_1 + 1 ]
4.17/2.21	
4.17/2.21	strictly and produces the following problem:
4.17/2.21	
4.17/2.21	4:	T:
4.17/2.21	
4.17/2.21			(Comp: 2*ar_0 + 2*ar_1, Cost: 1)    eval1(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
4.17/2.21	
4.17/2.21			(Comp: ar_1 + ar_2, Cost: 1)        eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 /\ ar_2 >= ar_1 + 1 ]
4.17/2.21	
4.17/2.21			(Comp: 2*ar_0 + 2*ar_1, Cost: 1)    eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0 - 1, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 /\ ar_1 >= ar_2 ]
4.17/2.21	
4.17/2.21			(Comp: 1, Cost: 1)                  start(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2))
4.17/2.21	
4.17/2.21			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2) -> Com_1(start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.17/2.21	
4.17/2.21		start location:	koat_start
4.17/2.21	
4.17/2.21		leaf cost:	0
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Complexity upper bound 4*ar_0 + 5*ar_1 + ar_2 + 1
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Time: 0.062 sec (SMT: 0.056 sec)
4.17/2.21	
4.17/2.21	
4.17/2.21	----------------------------------------
4.17/2.21	
4.17/2.21	(2)
4.17/2.21	BOUNDS(1, n^1)
4.17/2.21	
4.17/2.21	----------------------------------------
4.17/2.21	
4.17/2.21	(3) Loat Proof (FINISHED)
4.17/2.21	
4.17/2.21	
4.17/2.21	### Pre-processing the ITS problem ###
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Initial linear ITS problem
4.17/2.21	
4.17/2.21	   Start location: start
4.17/2.21	
4.17/2.21	      0: eval1 -> eval2 : [ A>=1+B ], cost: 1
4.17/2.21	
4.17/2.21	      1: eval2 -> eval2 : C'=-1+C, [ A>=1+B && C>=1+B ], cost: 1
4.17/2.21	
4.17/2.21	      2: eval2 -> eval1 : A'=-1+A, [ A>=1+B && B>=C ], cost: 1
4.17/2.21	
4.17/2.21	      3: start -> eval1 : [], cost: 1
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	### Simplification by acceleration and chaining ###
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Accelerating simple loops of location 1.
4.17/2.21	
4.17/2.21	   Accelerating the following rules:
4.17/2.21	
4.17/2.21	      1: eval2 -> eval2 : C'=-1+C, [ A>=1+B && C>=1+B ], cost: 1
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	   Accelerated rule 1 with metering function C-B, yielding the new rule 4.
4.17/2.21	
4.17/2.21	   Removing the simple loops: 1.
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Accelerated all simple loops using metering functions (where possible):
4.17/2.21	
4.17/2.21	   Start location: start
4.17/2.21	
4.17/2.21	      0: eval1 -> eval2 : [ A>=1+B ], cost: 1
4.17/2.21	
4.17/2.21	      2: eval2 -> eval1 : A'=-1+A, [ A>=1+B && B>=C ], cost: 1
4.17/2.21	
4.17/2.21	      4: eval2 -> eval2 : C'=B, [ A>=1+B && C>=1+B ], cost: C-B
4.17/2.21	
4.17/2.21	      3: start -> eval1 : [], cost: 1
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Chained accelerated rules (with incoming rules):
4.17/2.21	
4.17/2.21	   Start location: start
4.17/2.21	
4.17/2.21	      0: eval1 -> eval2 : [ A>=1+B ], cost: 1
4.17/2.21	
4.17/2.21	      5: eval1 -> eval2 : C'=B, [ A>=1+B && C>=1+B ], cost: 1+C-B
4.17/2.21	
4.17/2.21	      2: eval2 -> eval1 : A'=-1+A, [ A>=1+B && B>=C ], cost: 1
4.17/2.21	
4.17/2.21	      3: start -> eval1 : [], cost: 1
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Eliminated locations (on tree-shaped paths):
4.17/2.21	
4.17/2.21	   Start location: start
4.17/2.21	
4.17/2.21	      6: eval1 -> eval1 : A'=-1+A, [ A>=1+B && B>=C ], cost: 2
4.17/2.21	
4.17/2.21	      7: eval1 -> eval1 : A'=-1+A, C'=B, [ A>=1+B && C>=1+B ], cost: 2+C-B
4.17/2.21	
4.17/2.21	      3: start -> eval1 : [], cost: 1
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Accelerating simple loops of location 0.
4.17/2.21	
4.17/2.21	   Accelerating the following rules:
4.17/2.21	
4.17/2.21	      6: eval1 -> eval1 : A'=-1+A, [ A>=1+B && B>=C ], cost: 2
4.17/2.21	
4.17/2.21	      7: eval1 -> eval1 : A'=-1+A, C'=B, [ A>=1+B && C>=1+B ], cost: 2+C-B
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	   Accelerated rule 6 with metering function A-B, yielding the new rule 8.
4.17/2.21	
4.17/2.21	   Found no metering function for rule 7.
4.17/2.21	
4.17/2.21	   Removing the simple loops: 6.
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Accelerated all simple loops using metering functions (where possible):
4.17/2.21	
4.17/2.21	   Start location: start
4.17/2.21	
4.17/2.21	      7: eval1 -> eval1 : A'=-1+A, C'=B, [ A>=1+B && C>=1+B ], cost: 2+C-B
4.17/2.21	
4.17/2.21	      8: eval1 -> eval1 : A'=B, [ A>=1+B && B>=C ], cost: 2*A-2*B
4.17/2.21	
4.17/2.21	      3: start -> eval1 : [], cost: 1
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Chained accelerated rules (with incoming rules):
4.17/2.21	
4.17/2.21	   Start location: start
4.17/2.21	
4.17/2.21	      3: start -> eval1 : [], cost: 1
4.17/2.21	
4.17/2.21	      9: start -> eval1 : A'=-1+A, C'=B, [ A>=1+B && C>=1+B ], cost: 3+C-B
4.17/2.21	
4.17/2.21	     10: start -> eval1 : A'=B, [ A>=1+B && B>=C ], cost: 1+2*A-2*B
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Removed unreachable locations (and leaf rules with constant cost):
4.17/2.21	
4.17/2.21	   Start location: start
4.17/2.21	
4.17/2.21	      9: start -> eval1 : A'=-1+A, C'=B, [ A>=1+B && C>=1+B ], cost: 3+C-B
4.17/2.21	
4.17/2.21	     10: start -> eval1 : A'=B, [ A>=1+B && B>=C ], cost: 1+2*A-2*B
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	### Computing asymptotic complexity ###
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Fully simplified ITS problem
4.17/2.21	
4.17/2.21	   Start location: start
4.17/2.21	
4.17/2.21	      9: start -> eval1 : A'=-1+A, C'=B, [ A>=1+B && C>=1+B ], cost: 3+C-B
4.17/2.21	
4.17/2.21	     10: start -> eval1 : A'=B, [ A>=1+B && B>=C ], cost: 1+2*A-2*B
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Computing asymptotic complexity for rule 9
4.17/2.21	
4.17/2.21	   Solved the limit problem by the following transformations:
4.17/2.21	
4.17/2.21	      Created initial limit problem:
4.17/2.21	
4.17/2.21	      3+C-B (+), C-B (+/+!), A-B (+/+!) [not solved]
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	      removing all constraints (solved by SMT)
4.17/2.21	
4.17/2.21	      resulting limit problem: [solved]
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	      applying transformation rule (C) using substitution {C==n,A==0,B==-1}
4.17/2.21	
4.17/2.21	      resulting limit problem:
4.17/2.21	
4.17/2.21	      [solved]
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	   Solution:
4.17/2.21	
4.17/2.21	      C / n
4.17/2.21	
4.17/2.21	      A / 0
4.17/2.21	
4.17/2.21	      B / -1
4.17/2.21	
4.17/2.21	   Resulting cost 4+n has complexity: Poly(n^1)
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Found new complexity Poly(n^1).
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	Obtained the following overall complexity (w.r.t. the length of the input n):
4.17/2.21	
4.17/2.21	   Complexity:  Poly(n^1)
4.17/2.21	
4.17/2.21	   Cpx degree:  1
4.17/2.21	
4.17/2.21	   Solved cost: 4+n
4.17/2.21	
4.17/2.21	   Rule cost:   3+C-B
4.17/2.21	
4.17/2.21	   Rule guard:  [ A>=1+B && C>=1+B ]
4.17/2.21	
4.17/2.21	
4.17/2.21	
4.17/2.21	WORST_CASE(Omega(n^1),?)
4.17/2.21	
4.17/2.21	
4.17/2.21	----------------------------------------
4.17/2.21	
4.17/2.21	(4)
4.17/2.21	BOUNDS(n^1, INF)
4.18/2.34	EOF