5.22/2.36	WORST_CASE(Omega(n^2), O(n^2))
5.22/2.37	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
5.22/2.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.22/2.37	
5.22/2.37	
5.22/2.37	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).
5.22/2.37	
5.22/2.37	(0) CpxIntTrs
5.22/2.37	(1) Koat Proof [FINISHED, 304 ms]
5.22/2.37	(2) BOUNDS(1, n^2)
5.22/2.37	(3) Loat Proof [FINISHED, 689 ms]
5.22/2.37	(4) BOUNDS(n^2, INF)
5.22/2.37	
5.22/2.37	
5.22/2.37	----------------------------------------
5.22/2.37	
5.22/2.37	(0)
5.22/2.37	Obligation:
5.22/2.37	Complexity Int TRS consisting of the following rules:
5.22/2.37	evalfstart(A, B, C, D) -> Com_1(evalfentryin(A, B, C, D)) :|: TRUE
5.22/2.37	evalfentryin(A, B, C, D) -> Com_1(evalfbb3in(0, 0, C, D)) :|: TRUE
5.22/2.37	evalfbb3in(A, B, C, D) -> Com_1(evalfbbin(A, B, C, D)) :|: C >= B + 1
5.22/2.37	evalfbb3in(A, B, C, D) -> Com_1(evalfreturnin(A, B, C, D)) :|: B >= C
5.22/2.37	evalfbbin(A, B, C, D) -> Com_1(evalfbb1in(A, B, C, D)) :|: D >= A + 1
5.22/2.37	evalfbbin(A, B, C, D) -> Com_1(evalfbb2in(A, B, C, D)) :|: A >= D
5.22/2.37	evalfbb1in(A, B, C, D) -> Com_1(evalfbb3in(A + 1, B, C, D)) :|: TRUE
5.22/2.37	evalfbb2in(A, B, C, D) -> Com_1(evalfbb3in(0, B + 1, C, D)) :|: TRUE
5.22/2.37	evalfreturnin(A, B, C, D) -> Com_1(evalfstop(A, B, C, D)) :|: TRUE
5.22/2.37	
5.22/2.37	The start-symbols are:[evalfstart_4]
5.22/2.37	
5.22/2.37	
5.22/2.37	----------------------------------------
5.22/2.37	
5.22/2.37	(1) Koat Proof (FINISHED)
5.22/2.37	YES(?, 6*ar_2 + 6*ar_3 + 12*ar_2*ar_3 + 7)
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Initial complexity problem:
5.22/2.37	
5.22/2.37	1:	T:
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, 0, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, ar_1 + 1, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
5.22/2.37	
5.22/2.37		start location:	koat_start
5.22/2.37	
5.22/2.37		leaf cost:	0
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Repeatedly propagating knowledge in problem 1 produces the following problem:
5.22/2.37	
5.22/2.37	2:	T:
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, 0, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, ar_1 + 1, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
5.22/2.37	
5.22/2.37		start location:	koat_start
5.22/2.37	
5.22/2.37		leaf cost:	0
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	A polynomial rank function with
5.22/2.37	
5.22/2.37		Pol(evalfstart) = 2
5.22/2.37	
5.22/2.37		Pol(evalfentryin) = 2
5.22/2.37	
5.22/2.37		Pol(evalfbb3in) = 2
5.22/2.37	
5.22/2.37		Pol(evalfbbin) = 2
5.22/2.37	
5.22/2.37		Pol(evalfreturnin) = 1
5.22/2.37	
5.22/2.37		Pol(evalfbb1in) = 2
5.22/2.37	
5.22/2.37		Pol(evalfbb2in) = 2
5.22/2.37	
5.22/2.37		Pol(evalfstop) = 0
5.22/2.37	
5.22/2.37		Pol(koat_start) = 2
5.22/2.37	
5.22/2.37	orients all transitions weakly and the transitions
5.22/2.37	
5.22/2.37		evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37		evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 ]
5.22/2.37	
5.22/2.37	strictly and produces the following problem:
5.22/2.37	
5.22/2.37	3:	T:
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, 0, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: 2, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, ar_1 + 1, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: 2, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
5.22/2.37	
5.22/2.37		start location:	koat_start
5.22/2.37	
5.22/2.37		leaf cost:	0
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Applied AI with 'oct' on problem 3 to obtain the following invariants:
5.22/2.37	
5.22/2.37	  For symbol evalfbb1in: X_4 - 1 >= 0 /\ X_3 + X_4 - 2 >= 0 /\ X_2 + X_4 - 1 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ -X_1 + X_4 - 1 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ -X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0
5.22/2.37	
5.22/2.37	  For symbol evalfbb2in: X_1 - X_4 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ -X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0
5.22/2.37	
5.22/2.37	  For symbol evalfbb3in: X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0
5.22/2.37	
5.22/2.37	  For symbol evalfbbin: X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ -X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0
5.22/2.37	
5.22/2.37	  For symbol evalfreturnin: X_2 - X_3 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	This yielded the following problem:
5.22/2.37	
5.22/2.37	4:	T:
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
5.22/2.37	
5.22/2.37			(Comp: 2, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_2 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, ar_1 + 1, ar_2, ar_3)) [ ar_0 - ar_3 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_3 - 1 >= 0 /\ ar_2 + ar_3 - 2 >= 0 /\ ar_1 + ar_3 - 1 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ -ar_0 + ar_3 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_3 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_0 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: 2, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_2 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_1 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, 0, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37		start location:	koat_start
5.22/2.37	
5.22/2.37		leaf cost:	0
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	A polynomial rank function with
5.22/2.37	
5.22/2.37		Pol(koat_start) = 2*V_3
5.22/2.37	
5.22/2.37		Pol(evalfstart) = 2*V_3
5.22/2.37	
5.22/2.37		Pol(evalfreturnin) = -2*V_2 + 2*V_3
5.22/2.37	
5.22/2.37		Pol(evalfstop) = -2*V_2 + 2*V_3
5.22/2.37	
5.22/2.37		Pol(evalfbb2in) = -2*V_2 + 2*V_3 - 1
5.22/2.37	
5.22/2.37		Pol(evalfbb3in) = -2*V_2 + 2*V_3
5.22/2.37	
5.22/2.37		Pol(evalfbb1in) = -2*V_2 + 2*V_3
5.22/2.37	
5.22/2.37		Pol(evalfbbin) = -2*V_2 + 2*V_3
5.22/2.37	
5.22/2.37		Pol(evalfentryin) = 2*V_3
5.22/2.37	
5.22/2.37	orients all transitions weakly and the transitions
5.22/2.37	
5.22/2.37		evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_3 ]
5.22/2.37	
5.22/2.37		evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, ar_1 + 1, ar_2, ar_3)) [ ar_0 - ar_3 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37	strictly and produces the following problem:
5.22/2.37	
5.22/2.37	5:	T:
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 0)         koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
5.22/2.37	
5.22/2.37			(Comp: 2, Cost: 1)         evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_2 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37			(Comp: 2*ar_2, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, ar_1 + 1, ar_2, ar_3)) [ ar_0 - ar_3 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)         evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_3 - 1 >= 0 /\ ar_2 + ar_3 - 2 >= 0 /\ ar_1 + ar_3 - 1 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ -ar_0 + ar_3 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37			(Comp: 2*ar_2, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_3 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)         evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_0 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: 2, Cost: 1)         evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_2 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)         evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_1 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 1)         evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, 0, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 1)         evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37		start location:	koat_start
5.22/2.37	
5.22/2.37		leaf cost:	0
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	A polynomial rank function with
5.22/2.37	
5.22/2.37		Pol(evalfbbin) = -2*V_1 + 2*V_4
5.22/2.37	
5.22/2.37		Pol(evalfbb1in) = -2*V_1 + 2*V_4 - 1
5.22/2.37	
5.22/2.37		Pol(evalfbb3in) = -2*V_1 + 2*V_4
5.22/2.37	
5.22/2.37	and size complexities
5.22/2.37	
5.22/2.37		S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0
5.22/2.37	
5.22/2.37		S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1
5.22/2.37	
5.22/2.37		S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2
5.22/2.37	
5.22/2.37		S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3
5.22/2.37	
5.22/2.37		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, 0, ar_2, ar_3))", 0-0) = 0
5.22/2.37	
5.22/2.37		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, 0, ar_2, ar_3))", 0-1) = 0
5.22/2.37	
5.22/2.37		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, 0, ar_2, ar_3))", 0-2) = ar_2
5.22/2.37	
5.22/2.37		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, 0, ar_2, ar_3))", 0-3) = ar_3
5.22/2.37	
5.22/2.37		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_2 >= ar_1 + 1 ]", 0-0) = ar_3
5.22/2.37	
5.22/2.37		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_2 >= ar_1 + 1 ]", 0-1) = ar_2
5.22/2.37	
5.22/2.37		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_2 >= ar_1 + 1 ]", 0-2) = ar_2
5.22/2.37	
5.22/2.37		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_2 >= ar_1 + 1 ]", 0-3) = ar_3
5.22/2.37	
5.22/2.37		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_1 >= ar_2 ]", 0-0) = 0
5.22/2.37	
5.22/2.37		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_1 >= ar_2 ]", 0-1) = ar_2
5.22/2.37	
5.22/2.37		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_1 >= ar_2 ]", 0-2) = ar_2
5.22/2.37	
5.22/2.37		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_1 >= ar_2 ]", 0-3) = ar_3
5.22/2.37	
5.22/2.37		S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_3 >= ar_0 + 1 ]", 0-0) = ar_3
5.22/2.37	
5.22/2.37		S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_3 >= ar_0 + 1 ]", 0-1) = ar_2
5.22/2.37	
5.22/2.37		S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_3 >= ar_0 + 1 ]", 0-2) = ar_2
5.22/2.37	
5.22/2.37		S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_3 >= ar_0 + 1 ]", 0-3) = ar_3
5.22/2.37	
5.22/2.37		S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_0 >= ar_3 ]", 0-0) = ar_3
5.22/2.37	
5.22/2.37		S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_0 >= ar_3 ]", 0-1) = ar_2
5.22/2.37	
5.22/2.37		S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_0 >= ar_3 ]", 0-2) = ar_2
5.22/2.37	
5.22/2.37		S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_0 >= ar_3 ]", 0-3) = ar_3
5.22/2.37	
5.22/2.37		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_3 - 1 >= 0 /\\ ar_2 + ar_3 - 2 >= 0 /\\ ar_1 + ar_3 - 1 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ -ar_0 + ar_3 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-0) = ar_3
5.22/2.37	
5.22/2.37		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_3 - 1 >= 0 /\\ ar_2 + ar_3 - 2 >= 0 /\\ ar_1 + ar_3 - 1 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ -ar_0 + ar_3 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-1) = ar_2
5.22/2.37	
5.22/2.37		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_3 - 1 >= 0 /\\ ar_2 + ar_3 - 2 >= 0 /\\ ar_1 + ar_3 - 1 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ -ar_0 + ar_3 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-2) = ar_2
5.22/2.37	
5.22/2.37		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_3 - 1 >= 0 /\\ ar_2 + ar_3 - 2 >= 0 /\\ ar_1 + ar_3 - 1 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ -ar_0 + ar_3 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-3) = ar_3
5.22/2.37	
5.22/2.37		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, ar_1 + 1, ar_2, ar_3)) [ ar_0 - ar_3 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-0) = 0
5.22/2.37	
5.22/2.37		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, ar_1 + 1, ar_2, ar_3)) [ ar_0 - ar_3 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-1) = ar_2
5.22/2.37	
5.22/2.37		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, ar_1 + 1, ar_2, ar_3)) [ ar_0 - ar_3 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-2) = ar_2
5.22/2.37	
5.22/2.37		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, ar_1 + 1, ar_2, ar_3)) [ ar_0 - ar_3 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-3) = ar_3
5.22/2.37	
5.22/2.37		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_2 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-0) = 0
5.22/2.37	
5.22/2.37		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_2 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-1) = ar_2
5.22/2.37	
5.22/2.37		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_2 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-2) = ar_2
5.22/2.37	
5.22/2.37		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_2 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-3) = ar_3
5.22/2.37	
5.22/2.37		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-0) = ar_0
5.22/2.37	
5.22/2.37		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-1) = ar_1
5.22/2.37	
5.22/2.37		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-2) = ar_2
5.22/2.37	
5.22/2.37		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3
5.22/2.37	
5.22/2.37	orients the transitions
5.22/2.37	
5.22/2.37		evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_0 + 1 ]
5.22/2.37	
5.22/2.37		evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_1 + 1 ]
5.22/2.37	
5.22/2.37		evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_3 - 1 >= 0 /\ ar_2 + ar_3 - 2 >= 0 /\ ar_1 + ar_3 - 1 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ -ar_0 + ar_3 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37	weakly and the transitions
5.22/2.37	
5.22/2.37		evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_0 + 1 ]
5.22/2.37	
5.22/2.37		evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_3 - 1 >= 0 /\ ar_2 + ar_3 - 2 >= 0 /\ ar_1 + ar_3 - 1 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ -ar_0 + ar_3 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37	strictly and produces the following problem:
5.22/2.37	
5.22/2.37	6:	T:
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 0)                       koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
5.22/2.37	
5.22/2.37			(Comp: 2, Cost: 1)                       evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_2 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37			(Comp: 2*ar_2, Cost: 1)                  evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, ar_1 + 1, ar_2, ar_3)) [ ar_0 - ar_3 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37			(Comp: 2*ar_3 + 4*ar_2*ar_3, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_3 - 1 >= 0 /\ ar_2 + ar_3 - 2 >= 0 /\ ar_1 + ar_3 - 1 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ -ar_0 + ar_3 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37			(Comp: 2*ar_2, Cost: 1)                  evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_3 ]
5.22/2.37	
5.22/2.37			(Comp: 2*ar_3 + 4*ar_2*ar_3, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_0 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: 2, Cost: 1)                       evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_2 ]
5.22/2.37	
5.22/2.37			(Comp: ?, Cost: 1)                       evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_1 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 1)                       evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, 0, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 1)                       evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37		start location:	koat_start
5.22/2.37	
5.22/2.37		leaf cost:	0
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Repeatedly propagating knowledge in problem 6 produces the following problem:
5.22/2.37	
5.22/2.37	7:	T:
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 0)                                    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
5.22/2.37	
5.22/2.37			(Comp: 2, Cost: 1)                                    evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_2 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37			(Comp: 2*ar_2, Cost: 1)                               evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, ar_1 + 1, ar_2, ar_3)) [ ar_0 - ar_3 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37			(Comp: 2*ar_3 + 4*ar_2*ar_3, Cost: 1)                 evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_3 - 1 >= 0 /\ ar_2 + ar_3 - 2 >= 0 /\ ar_1 + ar_3 - 1 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ -ar_0 + ar_3 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
5.22/2.37	
5.22/2.37			(Comp: 2*ar_2, Cost: 1)                               evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_3 ]
5.22/2.37	
5.22/2.37			(Comp: 2*ar_3 + 4*ar_2*ar_3, Cost: 1)                 evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_0 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: 2, Cost: 1)                                    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_2 ]
5.22/2.37	
5.22/2.37			(Comp: 2*ar_3 + 4*ar_2*ar_3 + 2*ar_2 + 1, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_1 + 1 ]
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 1)                                    evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(0, 0, ar_2, ar_3))
5.22/2.37	
5.22/2.37			(Comp: 1, Cost: 1)                                    evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
5.22/2.37	
5.22/2.37		start location:	koat_start
5.22/2.37	
5.22/2.37		leaf cost:	0
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Complexity upper bound 6*ar_2 + 6*ar_3 + 12*ar_2*ar_3 + 7
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Time: 0.295 sec (SMT: 0.247 sec)
5.22/2.37	
5.22/2.37	
5.22/2.37	----------------------------------------
5.22/2.37	
5.22/2.37	(2)
5.22/2.37	BOUNDS(1, n^2)
5.22/2.37	
5.22/2.37	----------------------------------------
5.22/2.37	
5.22/2.37	(3) Loat Proof (FINISHED)
5.22/2.37	
5.22/2.37	
5.22/2.37	### Pre-processing the ITS problem ###
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Initial linear ITS problem
5.22/2.37	
5.22/2.37	   Start location: evalfstart
5.22/2.37	
5.22/2.37	      0: evalfstart -> evalfentryin : [], cost: 1
5.22/2.37	
5.22/2.37	      1: evalfentryin -> evalfbb3in : A'=0, B'=0, [], cost: 1
5.22/2.37	
5.22/2.37	      2: evalfbb3in -> evalfbbin : [ C>=1+B ], cost: 1
5.22/2.37	
5.22/2.37	      3: evalfbb3in -> evalfreturnin : [ B>=C ], cost: 1
5.22/2.37	
5.22/2.37	      4: evalfbbin -> evalfbb1in : [ D>=1+A ], cost: 1
5.22/2.37	
5.22/2.37	      5: evalfbbin -> evalfbb2in : [ A>=D ], cost: 1
5.22/2.37	
5.22/2.37	      6: evalfbb1in -> evalfbb3in : A'=1+A, [], cost: 1
5.22/2.37	
5.22/2.37	      7: evalfbb2in -> evalfbb3in : A'=0, B'=1+B, [], cost: 1
5.22/2.37	
5.22/2.37	      8: evalfreturnin -> evalfstop : [], cost: 1
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Removed unreachable and leaf rules:
5.22/2.37	
5.22/2.37	   Start location: evalfstart
5.22/2.37	
5.22/2.37	      0: evalfstart -> evalfentryin : [], cost: 1
5.22/2.37	
5.22/2.37	      1: evalfentryin -> evalfbb3in : A'=0, B'=0, [], cost: 1
5.22/2.37	
5.22/2.37	      2: evalfbb3in -> evalfbbin : [ C>=1+B ], cost: 1
5.22/2.37	
5.22/2.37	      4: evalfbbin -> evalfbb1in : [ D>=1+A ], cost: 1
5.22/2.37	
5.22/2.37	      5: evalfbbin -> evalfbb2in : [ A>=D ], cost: 1
5.22/2.37	
5.22/2.37	      6: evalfbb1in -> evalfbb3in : A'=1+A, [], cost: 1
5.22/2.37	
5.22/2.37	      7: evalfbb2in -> evalfbb3in : A'=0, B'=1+B, [], cost: 1
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	### Simplification by acceleration and chaining ###
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Eliminated locations (on linear paths):
5.22/2.37	
5.22/2.37	   Start location: evalfstart
5.22/2.37	
5.22/2.37	      9: evalfstart -> evalfbb3in : A'=0, B'=0, [], cost: 2
5.22/2.37	
5.22/2.37	      2: evalfbb3in -> evalfbbin : [ C>=1+B ], cost: 1
5.22/2.37	
5.22/2.37	     10: evalfbbin -> evalfbb3in : A'=1+A, [ D>=1+A ], cost: 2
5.22/2.37	
5.22/2.37	     11: evalfbbin -> evalfbb3in : A'=0, B'=1+B, [ A>=D ], cost: 2
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Eliminated locations (on tree-shaped paths):
5.22/2.37	
5.22/2.37	   Start location: evalfstart
5.22/2.37	
5.22/2.37	      9: evalfstart -> evalfbb3in : A'=0, B'=0, [], cost: 2
5.22/2.37	
5.22/2.37	     12: evalfbb3in -> evalfbb3in : A'=1+A, [ C>=1+B && D>=1+A ], cost: 3
5.22/2.37	
5.22/2.37	     13: evalfbb3in -> evalfbb3in : A'=0, B'=1+B, [ C>=1+B && A>=D ], cost: 3
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Accelerating simple loops of location 2.
5.22/2.37	
5.22/2.37	   Accelerating the following rules:
5.22/2.37	
5.22/2.37	     12: evalfbb3in -> evalfbb3in : A'=1+A, [ C>=1+B && D>=1+A ], cost: 3
5.22/2.37	
5.22/2.37	     13: evalfbb3in -> evalfbb3in : A'=0, B'=1+B, [ C>=1+B && A>=D ], cost: 3
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	   Accelerated rule 12 with metering function D-A, yielding the new rule 14.
5.22/2.37	
5.22/2.37	   Accelerated rule 13 with metering function C-B (after strengthening guard), yielding the new rule 15.
5.22/2.37	
5.22/2.37	   Nested simple loops 13 (outer loop) and 14 (inner loop) with metering function -1+C-B, resulting in the new rules: 16, 17.
5.22/2.37	
5.22/2.37	   Removing the simple loops: 12 13.
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Accelerated all simple loops using metering functions (where possible):
5.22/2.37	
5.22/2.37	   Start location: evalfstart
5.22/2.37	
5.22/2.37	      9: evalfstart -> evalfbb3in : A'=0, B'=0, [], cost: 2
5.22/2.37	
5.22/2.37	     14: evalfbb3in -> evalfbb3in : A'=D, [ C>=1+B && D>=1+A ], cost: 3*D-3*A
5.22/2.37	
5.22/2.37	     15: evalfbb3in -> evalfbb3in : A'=0, B'=C, [ C>=1+B && A>=D && 0>=D ], cost: 3*C-3*B
5.22/2.37	
5.22/2.37	     16: evalfbb3in -> evalfbb3in : A'=D, B'=-1+C, [ A>=D && C>=2+B && D>=1 ], cost: -3+3*C+3*D*(-1+C-B)-3*B
5.22/2.37	
5.22/2.37	     17: evalfbb3in -> evalfbb3in : A'=D, B'=-1+C, [ D>=1+A && C>=2+B && D>=1 ], cost: -3+3*C+3*D-3*A+3*D*(-1+C-B)-3*B
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Chained accelerated rules (with incoming rules):
5.22/2.37	
5.22/2.37	   Start location: evalfstart
5.22/2.37	
5.22/2.37	      9: evalfstart -> evalfbb3in : A'=0, B'=0, [], cost: 2
5.22/2.37	
5.22/2.37	     18: evalfstart -> evalfbb3in : A'=D, B'=0, [ C>=1 && D>=1 ], cost: 2+3*D
5.22/2.37	
5.22/2.37	     19: evalfstart -> evalfbb3in : A'=0, B'=C, [ C>=1 && 0>=D ], cost: 2+3*C
5.22/2.37	
5.22/2.37	     20: evalfstart -> evalfbb3in : A'=D, B'=-1+C, [ D>=1 && C>=2 ], cost: -1+3*D*(-1+C)+3*C+3*D
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Removed unreachable locations (and leaf rules with constant cost):
5.22/2.37	
5.22/2.37	   Start location: evalfstart
5.22/2.37	
5.22/2.37	     18: evalfstart -> evalfbb3in : A'=D, B'=0, [ C>=1 && D>=1 ], cost: 2+3*D
5.22/2.37	
5.22/2.37	     19: evalfstart -> evalfbb3in : A'=0, B'=C, [ C>=1 && 0>=D ], cost: 2+3*C
5.22/2.37	
5.22/2.37	     20: evalfstart -> evalfbb3in : A'=D, B'=-1+C, [ D>=1 && C>=2 ], cost: -1+3*D*(-1+C)+3*C+3*D
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	### Computing asymptotic complexity ###
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Fully simplified ITS problem
5.22/2.37	
5.22/2.37	   Start location: evalfstart
5.22/2.37	
5.22/2.37	     18: evalfstart -> evalfbb3in : A'=D, B'=0, [ C>=1 && D>=1 ], cost: 2+3*D
5.22/2.37	
5.22/2.37	     19: evalfstart -> evalfbb3in : A'=0, B'=C, [ C>=1 && 0>=D ], cost: 2+3*C
5.22/2.37	
5.22/2.37	     20: evalfstart -> evalfbb3in : A'=D, B'=-1+C, [ D>=1 && C>=2 ], cost: -1+3*D*(-1+C)+3*C+3*D
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Computing asymptotic complexity for rule 18
5.22/2.37	
5.22/2.37	   Solved the limit problem by the following transformations:
5.22/2.37	
5.22/2.37	      Created initial limit problem:
5.22/2.37	
5.22/2.37	      C (+/+!), D (+/+!), 2+3*D (+) [not solved]
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	      removing all constraints (solved by SMT)
5.22/2.37	
5.22/2.37	      resulting limit problem: [solved]
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	      applying transformation rule (C) using substitution {C==1,D==n}
5.22/2.37	
5.22/2.37	      resulting limit problem:
5.22/2.37	
5.22/2.37	      [solved]
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	   Solution:
5.22/2.37	
5.22/2.37	      C / 1
5.22/2.37	
5.22/2.37	      D / n
5.22/2.37	
5.22/2.37	   Resulting cost 2+3*n has complexity: Poly(n^1)
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Found new complexity Poly(n^1).
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Computing asymptotic complexity for rule 20
5.22/2.37	
5.22/2.37	   Solved the limit problem by the following transformations:
5.22/2.37	
5.22/2.37	      Created initial limit problem:
5.22/2.37	
5.22/2.37	      -1+3*C+3*C*D (+), D (+/+!), -1+C (+/+!) [not solved]
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	      removing all constraints (solved by SMT)
5.22/2.37	
5.22/2.37	      resulting limit problem: [solved]
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	      applying transformation rule (C) using substitution {C==n,D==n}
5.22/2.37	
5.22/2.37	      resulting limit problem:
5.22/2.37	
5.22/2.37	      [solved]
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	   Solution:
5.22/2.37	
5.22/2.37	      C / n
5.22/2.37	
5.22/2.37	      D / n
5.22/2.37	
5.22/2.37	   Resulting cost -1+3*n^2+3*n has complexity: Poly(n^2)
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Found new complexity Poly(n^2).
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	Obtained the following overall complexity (w.r.t. the length of the input n):
5.22/2.37	
5.22/2.37	   Complexity:  Poly(n^2)
5.22/2.37	
5.22/2.37	   Cpx degree:  2
5.22/2.37	
5.22/2.37	   Solved cost: -1+3*n^2+3*n
5.22/2.37	
5.22/2.37	   Rule cost:   -1+3*D*(-1+C)+3*C+3*D
5.22/2.37	
5.22/2.37	   Rule guard:  [ D>=1 && C>=2 ]
5.22/2.37	
5.22/2.37	
5.22/2.37	
5.22/2.37	WORST_CASE(Omega(n^2),?)
5.22/2.37	
5.22/2.37	
5.22/2.37	----------------------------------------
5.22/2.37	
5.22/2.37	(4)
5.22/2.37	BOUNDS(n^2, INF)
5.22/2.38	EOF