4.76/2.19	WORST_CASE(Omega(n^2), O(n^2))
4.76/2.20	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.76/2.20	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.76/2.20	
4.76/2.20	
4.76/2.20	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).
4.76/2.20	
4.76/2.20	(0) CpxIntTrs
4.76/2.20	(1) Koat Proof [FINISHED, 118 ms]
4.76/2.20	(2) BOUNDS(1, n^2)
4.76/2.20	(3) Loat Proof [FINISHED, 589 ms]
4.76/2.20	(4) BOUNDS(n^2, INF)
4.76/2.20	
4.76/2.20	
4.76/2.20	----------------------------------------
4.76/2.20	
4.76/2.20	(0)
4.76/2.20	Obligation:
4.76/2.20	Complexity Int TRS consisting of the following rules:
4.76/2.20	evalfstart(A, B, C, D) -> Com_1(evalfentryin(A, B, C, D)) :|: TRUE
4.76/2.20	evalfentryin(A, B, C, D) -> Com_1(evalfbb4in(1, B, C, D)) :|: TRUE
4.76/2.20	evalfbb4in(A, B, C, D) -> Com_1(evalfbb2in(A, B, 1, D)) :|: B >= A
4.76/2.20	evalfbb4in(A, B, C, D) -> Com_1(evalfreturnin(A, B, C, D)) :|: A >= B + 1
4.76/2.20	evalfbb2in(A, B, C, D) -> Com_1(evalfbb1in(A, B, C, D)) :|: D >= C
4.76/2.20	evalfbb2in(A, B, C, D) -> Com_1(evalfbb3in(A, B, C, D)) :|: C >= D + 1
4.76/2.20	evalfbb1in(A, B, C, D) -> Com_1(evalfbb2in(A, B, C + 1, D)) :|: TRUE
4.76/2.20	evalfbb3in(A, B, C, D) -> Com_1(evalfbb4in(A + 1, B, C, D)) :|: TRUE
4.76/2.20	evalfreturnin(A, B, C, D) -> Com_1(evalfstop(A, B, C, D)) :|: TRUE
4.76/2.20	
4.76/2.20	The start-symbols are:[evalfstart_4]
4.76/2.20	
4.76/2.20	
4.76/2.20	----------------------------------------
4.76/2.20	
4.76/2.20	(1) Koat Proof (FINISHED)
4.76/2.20	YES(?, 9*ar_1 + 2*ar_1*ar_3 + 6)
4.76/2.20	
4.76/2.20	
4.76/2.20	
4.76/2.20	Initial complexity problem:
4.76/2.20	
4.76/2.20	1:	T:
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(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 ]
4.76/2.20	
4.76/2.20		start location:	koat_start
4.76/2.20	
4.76/2.20		leaf cost:	0
4.76/2.20	
4.76/2.20	
4.76/2.20	
4.76/2.20	Repeatedly propagating knowledge in problem 1 produces the following problem:
4.76/2.20	
4.76/2.20	2:	T:
4.76/2.20	
4.76/2.20			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(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 ]
4.76/2.20	
4.76/2.20		start location:	koat_start
4.76/2.20	
4.76/2.20		leaf cost:	0
4.76/2.20	
4.76/2.20	
4.76/2.20	
4.76/2.20	A polynomial rank function with
4.76/2.20	
4.76/2.20		Pol(evalfstart) = 2
4.76/2.20	
4.76/2.20		Pol(evalfentryin) = 2
4.76/2.20	
4.76/2.20		Pol(evalfbb4in) = 2
4.76/2.20	
4.76/2.20		Pol(evalfbb2in) = 2
4.76/2.20	
4.76/2.20		Pol(evalfreturnin) = 1
4.76/2.20	
4.76/2.20		Pol(evalfbb1in) = 2
4.76/2.20	
4.76/2.20		Pol(evalfbb3in) = 2
4.76/2.20	
4.76/2.20		Pol(evalfstop) = 0
4.76/2.20	
4.76/2.20		Pol(koat_start) = 2
4.76/2.20	
4.76/2.20	orients all transitions weakly and the transitions
4.76/2.20	
4.76/2.20		evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20		evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
4.76/2.20	
4.76/2.20	strictly and produces the following problem:
4.76/2.20	
4.76/2.20	3:	T:
4.76/2.20	
4.76/2.20			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
4.76/2.20	
4.76/2.20			(Comp: 2, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(Comp: 2, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(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 ]
4.76/2.20	
4.76/2.20		start location:	koat_start
4.76/2.20	
4.76/2.20		leaf cost:	0
4.76/2.20	
4.76/2.20	
4.76/2.20	
4.76/2.20	A polynomial rank function with
4.76/2.20	
4.76/2.20		Pol(evalfstart) = V_2
4.76/2.20	
4.76/2.20		Pol(evalfentryin) = V_2
4.76/2.20	
4.76/2.20		Pol(evalfbb4in) = -V_1 + V_2 + 1
4.76/2.20	
4.76/2.20		Pol(evalfbb2in) = -V_1 + V_2
4.76/2.20	
4.76/2.20		Pol(evalfreturnin) = -V_1 + V_2
4.76/2.20	
4.76/2.20		Pol(evalfbb1in) = -V_1 + V_2
4.76/2.20	
4.76/2.20		Pol(evalfbb3in) = -V_1 + V_2
4.76/2.20	
4.76/2.20		Pol(evalfstop) = -V_1 + V_2
4.76/2.20	
4.76/2.20		Pol(koat_start) = V_2
4.76/2.20	
4.76/2.20	orients all transitions weakly and the transition
4.76/2.20	
4.76/2.20		evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
4.76/2.20	
4.76/2.20	strictly and produces the following problem:
4.76/2.20	
4.76/2.20	4:	T:
4.76/2.20	
4.76/2.20			(Comp: 1, Cost: 1)       evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(Comp: 1, Cost: 1)       evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(Comp: ar_1, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
4.76/2.20	
4.76/2.20			(Comp: 2, Cost: 1)       evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)       evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)       evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)       evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
4.76/2.20	
4.76/2.20			(Comp: ?, Cost: 1)       evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(Comp: 2, Cost: 1)       evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20			(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 ]
4.76/2.20	
4.76/2.20		start location:	koat_start
4.76/2.20	
4.76/2.20		leaf cost:	0
4.76/2.20	
4.76/2.20	
4.76/2.20	
4.76/2.20	A polynomial rank function with
4.76/2.20	
4.76/2.20		Pol(evalfbb3in) = 1
4.76/2.20	
4.76/2.20		Pol(evalfbb4in) = 0
4.76/2.20	
4.76/2.20		Pol(evalfbb2in) = 2
4.76/2.20	
4.76/2.20		Pol(evalfbb1in) = 2
4.76/2.20	
4.76/2.20	and size complexities
4.76/2.20	
4.76/2.20		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
4.76/2.20	
4.76/2.20		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
4.76/2.20	
4.76/2.20		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
4.76/2.20	
4.76/2.20		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
4.76/2.20	
4.76/2.20		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-0) = ?
4.76/2.20	
4.76/2.20		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1
4.76/2.20	
4.76/2.20		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-2) = ?
4.76/2.20	
4.76/2.20		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3
4.76/2.20	
4.76/2.20		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-0) = ?
4.76/2.20	
4.76/2.20		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-1) = ar_1
4.76/2.20	
4.76/2.20		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-2) = ?
4.76/2.20	
4.76/2.20		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-3) = ar_3
4.76/2.20	
4.76/2.20		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-0) = ?
4.76/2.20	
4.76/2.20		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-1) = ar_1
4.76/2.20	
4.76/2.20		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-2) = ?
4.76/2.20	
4.76/2.20		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-3) = ar_3
4.76/2.20	
4.76/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-0) = ?
4.76/2.20	
4.76/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-1) = ar_1
4.76/2.20	
4.76/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-2) = ?
4.76/2.20	
4.76/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-3) = ar_3
4.76/2.20	
4.76/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-0) = ?
4.76/2.20	
4.76/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-1) = ar_1
4.76/2.20	
4.76/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-2) = ?
4.76/2.20	
4.76/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-3) = ar_3
4.76/2.20	
4.76/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-0) = ?
4.76/2.20	
4.76/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ar_1
4.76/2.20	
4.76/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-2) = ?
4.76/2.20	
4.76/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-3) = ar_3
4.76/2.20	
4.76/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-0) = ?
4.76/2.20	
4.76/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-1) = ar_1
4.76/2.20	
4.76/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-2) = 1
4.76/2.20	
4.76/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-3) = ar_3
4.76/2.20	
4.76/2.20		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-0) = 1
4.76/2.20	
4.76/2.20		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-1) = ar_1
4.76/2.20	
4.76/2.20		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-2) = ar_2
4.76/2.20	
4.76/2.20		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-3) = ar_3
4.76/2.20	
4.76/2.20		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
4.76/2.20	
4.76/2.20		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
4.76/2.20	
4.76/2.20		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
4.76/2.20	
4.76/2.20		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
4.76/2.20	
4.76/2.20	orients the transitions
4.76/2.20	
4.76/2.20		evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20		evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
4.76/2.20	
4.76/2.20		evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
4.76/2.20	
4.76/2.20		evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
4.76/2.20	
4.76/2.20	weakly and the transitions
4.76/2.20	
4.76/2.20		evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
4.76/2.20	
4.76/2.20		evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
4.76/2.20	
4.76/2.20	strictly and produces the following problem:
4.88/2.20	
4.88/2.20	5:	T:
4.88/2.20	
4.88/2.20			(Comp: 1, Cost: 1)         evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
4.88/2.20	
4.88/2.20			(Comp: 1, Cost: 1)         evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
4.88/2.20	
4.88/2.20			(Comp: ar_1, Cost: 1)      evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
4.88/2.20	
4.88/2.20			(Comp: 2, Cost: 1)         evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
4.88/2.20	
4.88/2.20			(Comp: ?, Cost: 1)         evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
4.88/2.20	
4.88/2.20			(Comp: 2*ar_1, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
4.88/2.20	
4.88/2.20			(Comp: ?, Cost: 1)         evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
4.88/2.20	
4.88/2.20			(Comp: 2*ar_1, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
4.88/2.20	
4.88/2.20			(Comp: 2, Cost: 1)         evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
4.88/2.20	
4.88/2.20			(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 ]
4.88/2.20	
4.88/2.20		start location:	koat_start
4.88/2.20	
4.88/2.20		leaf cost:	0
4.88/2.20	
4.88/2.20	
4.88/2.20	
4.88/2.20	A polynomial rank function with
4.88/2.20	
4.88/2.20		Pol(evalfbb2in) = -V_3 + V_4 + 1
4.88/2.20	
4.88/2.20		Pol(evalfbb1in) = -V_3 + V_4
4.88/2.20	
4.88/2.20	and size complexities
4.88/2.20	
4.88/2.20		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
4.88/2.20	
4.88/2.20		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
4.88/2.20	
4.88/2.20		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
4.88/2.20	
4.88/2.20		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
4.88/2.20	
4.88/2.20		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-0) = 2*ar_1 + 20
4.88/2.20	
4.88/2.20		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1
4.88/2.20	
4.88/2.20		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-2) = ?
4.88/2.20	
4.88/2.20		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3
4.88/2.20	
4.88/2.20		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-0) = 2*ar_1 + 4
4.88/2.20	
4.88/2.20		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-1) = ar_1
4.88/2.20	
4.88/2.20		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-2) = ?
4.88/2.20	
4.88/2.20		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-3) = ar_3
4.88/2.20	
4.88/2.20		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-0) = 2*ar_1 + 4
4.88/2.20	
4.88/2.20		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-1) = ar_1
4.88/2.20	
4.88/2.20		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-2) = ?
4.88/2.20	
4.88/2.20		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-3) = ar_3
4.88/2.20	
4.88/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-0) = 2*ar_1 + 4
4.88/2.20	
4.88/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-1) = ar_1
4.88/2.20	
4.88/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-2) = ?
4.88/2.20	
4.88/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-3) = ar_3
4.88/2.20	
4.88/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-0) = 2*ar_1 + 4
4.88/2.20	
4.88/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-1) = ar_1
4.88/2.20	
4.88/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-2) = ?
4.88/2.20	
4.88/2.20		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-3) = ar_3
4.88/2.20	
4.88/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-0) = 2*ar_1 + 10
4.88/2.20	
4.88/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ar_1
4.88/2.20	
4.88/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-2) = ?
4.88/2.20	
4.88/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-3) = ar_3
4.88/2.20	
4.88/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-0) = 2*ar_1 + 4
4.88/2.20	
4.88/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-1) = ar_1
4.88/2.20	
4.88/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-2) = 1
4.88/2.20	
4.88/2.20		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-3) = ar_3
4.88/2.20	
4.88/2.20		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-0) = 1
4.88/2.20	
4.88/2.20		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-1) = ar_1
4.88/2.20	
4.88/2.20		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-2) = ar_2
4.88/2.20	
4.88/2.20		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-3) = ar_3
4.88/2.20	
4.88/2.20		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
4.88/2.20	
4.88/2.20		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
4.88/2.20	
4.88/2.20		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
4.88/2.20	
4.88/2.20		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
4.88/2.20	
4.88/2.20	orients the transitions
4.88/2.20	
4.88/2.20		evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
4.88/2.20	
4.88/2.20		evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
4.88/2.20	
4.88/2.20	weakly and the transition
4.88/2.20	
4.88/2.20		evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
4.88/2.20	
4.88/2.20	strictly and produces the following problem:
4.88/2.20	
4.88/2.20	6:	T:
4.88/2.20	
4.88/2.20			(Comp: 1, Cost: 1)                     evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
4.88/2.20	
4.88/2.20			(Comp: 1, Cost: 1)                     evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
4.88/2.20	
4.88/2.20			(Comp: ar_1, Cost: 1)                  evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
4.88/2.20	
4.88/2.20			(Comp: 2, Cost: 1)                     evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
4.88/2.20	
4.88/2.20			(Comp: ar_1*ar_3 + 2*ar_1, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
4.88/2.20	
4.88/2.20			(Comp: 2*ar_1, Cost: 1)                evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
4.88/2.20	
4.88/2.20			(Comp: ?, Cost: 1)                     evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
4.88/2.20	
4.88/2.20			(Comp: 2*ar_1, Cost: 1)                evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
4.88/2.20	
4.88/2.20			(Comp: 2, Cost: 1)                     evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
4.88/2.20	
4.88/2.20			(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 ]
4.88/2.20	
4.88/2.20		start location:	koat_start
4.88/2.20	
4.88/2.20		leaf cost:	0
4.88/2.20	
4.88/2.20	
4.88/2.20	
4.88/2.20	Repeatedly propagating knowledge in problem 6 produces the following problem:
4.88/2.20	
4.88/2.20	7:	T:
4.88/2.20	
4.88/2.20			(Comp: 1, Cost: 1)                     evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
4.88/2.20	
4.88/2.20			(Comp: 1, Cost: 1)                     evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
4.88/2.20	
4.88/2.20			(Comp: ar_1, Cost: 1)                  evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
4.88/2.20	
4.88/2.20			(Comp: 2, Cost: 1)                     evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
4.88/2.20	
4.88/2.20			(Comp: ar_1*ar_3 + 2*ar_1, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
4.88/2.20	
4.88/2.20			(Comp: 2*ar_1, Cost: 1)                evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
4.88/2.20	
4.88/2.20			(Comp: ar_1*ar_3 + 2*ar_1, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
4.88/2.20	
4.88/2.20			(Comp: 2*ar_1, Cost: 1)                evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
4.88/2.20	
4.88/2.20			(Comp: 2, Cost: 1)                     evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
4.88/2.20	
4.88/2.20			(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 ]
4.88/2.20	
4.88/2.20		start location:	koat_start
4.88/2.20	
4.88/2.20		leaf cost:	0
4.88/2.20	
4.88/2.20	
4.88/2.20	
4.88/2.20	Complexity upper bound 9*ar_1 + 2*ar_1*ar_3 + 6
4.88/2.20	
4.88/2.20	
4.88/2.20	
4.88/2.20	Time: 0.112 sec (SMT: 0.095 sec)
4.88/2.20	
4.88/2.20	
4.88/2.20	----------------------------------------
4.88/2.20	
4.88/2.20	(2)
4.88/2.20	BOUNDS(1, n^2)
4.88/2.20	
4.88/2.20	----------------------------------------
4.88/2.20	
4.88/2.20	(3) Loat Proof (FINISHED)
4.88/2.20	
4.88/2.20	
4.88/2.20	### Pre-processing the ITS problem ###
4.88/2.20	
4.88/2.20	
4.88/2.20	
4.88/2.20	Initial linear ITS problem
4.88/2.20	
4.88/2.20	   Start location: evalfstart
4.88/2.20	
4.88/2.20	      0: evalfstart -> evalfentryin : [], cost: 1
4.88/2.20	
4.88/2.20	      1: evalfentryin -> evalfbb4in : A'=1, [], cost: 1
4.88/2.20	
4.88/2.20	      2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1
4.88/2.20	
4.88/2.20	      3: evalfbb4in -> evalfreturnin : [ A>=1+B ], cost: 1
4.88/2.20	
4.88/2.20	      4: evalfbb2in -> evalfbb1in : [ D>=C ], cost: 1
4.88/2.20	
4.88/2.20	      5: evalfbb2in -> evalfbb3in : [ C>=1+D ], cost: 1
4.88/2.20	
4.88/2.20	      6: evalfbb1in -> evalfbb2in : C'=1+C, [], cost: 1
4.88/2.20	
4.88/2.20	      7: evalfbb3in -> evalfbb4in : A'=1+A, [], cost: 1
4.88/2.20	
4.88/2.20	      8: evalfreturnin -> evalfstop : [], cost: 1
4.88/2.20	
4.88/2.20	
4.88/2.20	
4.88/2.20	Removed unreachable and leaf rules:
4.88/2.20	
4.88/2.20	   Start location: evalfstart
4.88/2.20	
4.88/2.20	      0: evalfstart -> evalfentryin : [], cost: 1
4.88/2.20	
4.88/2.20	      1: evalfentryin -> evalfbb4in : A'=1, [], cost: 1
4.88/2.20	
4.88/2.20	      2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1
4.88/2.20	
4.88/2.20	      4: evalfbb2in -> evalfbb1in : [ D>=C ], cost: 1
4.88/2.20	
4.88/2.20	      5: evalfbb2in -> evalfbb3in : [ C>=1+D ], cost: 1
4.88/2.20	
4.88/2.20	      6: evalfbb1in -> evalfbb2in : C'=1+C, [], cost: 1
4.88/2.20	
4.88/2.20	      7: evalfbb3in -> evalfbb4in : A'=1+A, [], cost: 1
4.88/2.20	
4.88/2.20	
4.88/2.20	
4.88/2.20	### Simplification by acceleration and chaining ###
4.88/2.20	
4.88/2.20	
4.88/2.20	
4.88/2.20	Eliminated locations (on linear paths):
4.88/2.20	
4.88/2.20	   Start location: evalfstart
4.88/2.20	
4.88/2.20	      9: evalfstart -> evalfbb4in : A'=1, [], cost: 2
4.88/2.20	
4.88/2.20	      2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1
4.88/2.20	
4.88/2.20	     10: evalfbb2in -> evalfbb2in : C'=1+C, [ D>=C ], cost: 2
4.88/2.20	
4.88/2.20	     11: evalfbb2in -> evalfbb4in : A'=1+A, [ C>=1+D ], cost: 2
4.88/2.20	
4.88/2.20	
4.88/2.20	
4.88/2.20	Accelerating simple loops of location 3.
4.88/2.20	
4.88/2.20	   Accelerating the following rules:
4.88/2.20	
4.88/2.20	     10: evalfbb2in -> evalfbb2in : C'=1+C, [ D>=C ], cost: 2
4.88/2.20	
4.88/2.20	
4.88/2.20	
4.88/2.20	   Accelerated rule 10 with metering function 1-C+D, yielding the new rule 12.
4.88/2.20	
4.88/2.20	   Removing the simple loops: 10.
4.88/2.20	
4.88/2.20	
4.88/2.20	
4.88/2.20	Accelerated all simple loops using metering functions (where possible):
4.88/2.20	
4.88/2.20	   Start location: evalfstart
4.88/2.20	
4.88/2.20	      9: evalfstart -> evalfbb4in : A'=1, [], cost: 2
4.88/2.20	
4.88/2.20	      2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1
4.88/2.20	
4.88/2.20	     11: evalfbb2in -> evalfbb4in : A'=1+A, [ C>=1+D ], cost: 2
4.88/2.20	
4.88/2.20	     12: evalfbb2in -> evalfbb2in : C'=1+D, [ D>=C ], cost: 2-2*C+2*D
4.88/2.20	
4.88/2.20	
4.88/2.20	
4.88/2.20	Chained accelerated rules (with incoming rules):
4.88/2.21	
4.88/2.21	   Start location: evalfstart
4.88/2.21	
4.88/2.21	      9: evalfstart -> evalfbb4in : A'=1, [], cost: 2
4.88/2.21	
4.88/2.21	      2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1
4.88/2.21	
4.88/2.21	     13: evalfbb4in -> evalfbb2in : C'=1+D, [ B>=A && D>=1 ], cost: 1+2*D
4.88/2.21	
4.88/2.21	     11: evalfbb2in -> evalfbb4in : A'=1+A, [ C>=1+D ], cost: 2
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	Eliminated locations (on tree-shaped paths):
4.88/2.21	
4.88/2.21	   Start location: evalfstart
4.88/2.21	
4.88/2.21	      9: evalfstart -> evalfbb4in : A'=1, [], cost: 2
4.88/2.21	
4.88/2.21	     14: evalfbb4in -> evalfbb4in : A'=1+A, C'=1, [ B>=A && 1>=1+D ], cost: 3
4.88/2.21	
4.88/2.21	     15: evalfbb4in -> evalfbb4in : A'=1+A, C'=1+D, [ B>=A && D>=1 ], cost: 3+2*D
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	Accelerating simple loops of location 2.
4.88/2.21	
4.88/2.21	   Accelerating the following rules:
4.88/2.21	
4.88/2.21	     14: evalfbb4in -> evalfbb4in : A'=1+A, C'=1, [ B>=A && 1>=1+D ], cost: 3
4.88/2.21	
4.88/2.21	     15: evalfbb4in -> evalfbb4in : A'=1+A, C'=1+D, [ B>=A && D>=1 ], cost: 3+2*D
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	   Accelerated rule 14 with metering function 1-A+B, yielding the new rule 16.
4.88/2.21	
4.88/2.21	   Accelerated rule 15 with metering function 1-A+B, yielding the new rule 17.
4.88/2.21	
4.88/2.21	   Removing the simple loops: 14 15.
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	Accelerated all simple loops using metering functions (where possible):
4.88/2.21	
4.88/2.21	   Start location: evalfstart
4.88/2.21	
4.88/2.21	      9: evalfstart -> evalfbb4in : A'=1, [], cost: 2
4.88/2.21	
4.88/2.21	     16: evalfbb4in -> evalfbb4in : A'=1+B, C'=1, [ B>=A && 1>=1+D ], cost: 3-3*A+3*B
4.88/2.21	
4.88/2.21	     17: evalfbb4in -> evalfbb4in : A'=1+B, C'=1+D, [ B>=A && D>=1 ], cost: 3-2*(-1+A-B)*D-3*A+3*B
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	Chained accelerated rules (with incoming rules):
4.88/2.21	
4.88/2.21	   Start location: evalfstart
4.88/2.21	
4.88/2.21	      9: evalfstart -> evalfbb4in : A'=1, [], cost: 2
4.88/2.21	
4.88/2.21	     18: evalfstart -> evalfbb4in : A'=1+B, C'=1, [ B>=1 && 1>=1+D ], cost: 2+3*B
4.88/2.21	
4.88/2.21	     19: evalfstart -> evalfbb4in : A'=1+B, C'=1+D, [ B>=1 && D>=1 ], cost: 2+2*D*B+3*B
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	Removed unreachable locations (and leaf rules with constant cost):
4.88/2.21	
4.88/2.21	   Start location: evalfstart
4.88/2.21	
4.88/2.21	     18: evalfstart -> evalfbb4in : A'=1+B, C'=1, [ B>=1 && 1>=1+D ], cost: 2+3*B
4.88/2.21	
4.88/2.21	     19: evalfstart -> evalfbb4in : A'=1+B, C'=1+D, [ B>=1 && D>=1 ], cost: 2+2*D*B+3*B
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	### Computing asymptotic complexity ###
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	Fully simplified ITS problem
4.88/2.21	
4.88/2.21	   Start location: evalfstart
4.88/2.21	
4.88/2.21	     18: evalfstart -> evalfbb4in : A'=1+B, C'=1, [ B>=1 && 1>=1+D ], cost: 2+3*B
4.88/2.21	
4.88/2.21	     19: evalfstart -> evalfbb4in : A'=1+B, C'=1+D, [ B>=1 && D>=1 ], cost: 2+2*D*B+3*B
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	Computing asymptotic complexity for rule 18
4.88/2.21	
4.88/2.21	   Solved the limit problem by the following transformations:
4.88/2.21	
4.88/2.21	      Created initial limit problem:
4.88/2.21	
4.88/2.21	      1-D (+/+!), 2+3*B (+), B (+/+!) [not solved]
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	      removing all constraints (solved by SMT)
4.88/2.21	
4.88/2.21	      resulting limit problem: [solved]
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	      applying transformation rule (C) using substitution {D==0,B==n}
4.88/2.21	
4.88/2.21	      resulting limit problem:
4.88/2.21	
4.88/2.21	      [solved]
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	   Solution:
4.88/2.21	
4.88/2.21	      D / 0
4.88/2.21	
4.88/2.21	      B / n
4.88/2.21	
4.88/2.21	   Resulting cost 2+3*n has complexity: Poly(n^1)
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	Found new complexity Poly(n^1).
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	Computing asymptotic complexity for rule 19
4.88/2.21	
4.88/2.21	   Solved the limit problem by the following transformations:
4.88/2.21	
4.88/2.21	      Created initial limit problem:
4.88/2.21	
4.88/2.21	      2+2*D*B+3*B (+), D (+/+!), B (+/+!) [not solved]
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	      removing all constraints (solved by SMT)
4.88/2.21	
4.88/2.21	      resulting limit problem: [solved]
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	      applying transformation rule (C) using substitution {D==n,B==n}
4.88/2.21	
4.88/2.21	      resulting limit problem:
4.88/2.21	
4.88/2.21	      [solved]
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	   Solution:
4.88/2.21	
4.88/2.21	      D / n
4.88/2.21	
4.88/2.21	      B / n
4.88/2.21	
4.88/2.21	   Resulting cost 2+3*n+2*n^2 has complexity: Poly(n^2)
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	Found new complexity Poly(n^2).
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	Obtained the following overall complexity (w.r.t. the length of the input n):
4.88/2.21	
4.88/2.21	   Complexity:  Poly(n^2)
4.88/2.21	
4.88/2.21	   Cpx degree:  2
4.88/2.21	
4.88/2.21	   Solved cost: 2+3*n+2*n^2
4.88/2.21	
4.88/2.21	   Rule cost:   2+2*D*B+3*B
4.88/2.21	
4.88/2.21	   Rule guard:  [ B>=1 && D>=1 ]
4.88/2.21	
4.88/2.21	
4.88/2.21	
4.88/2.21	WORST_CASE(Omega(n^2),?)
4.88/2.21	
4.88/2.21	
4.88/2.21	----------------------------------------
4.88/2.21	
4.88/2.21	(4)
4.88/2.21	BOUNDS(n^2, INF)
4.88/2.23	EOF