2.09/1.31	WORST_CASE(?, O(n^2))
2.09/1.32	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.09/1.32	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.09/1.32	
2.09/1.32	
2.09/1.32	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2).
2.09/1.32	
2.09/1.32	(0) CpxIntTrs
2.09/1.32	(1) Koat Proof [FINISHED, 79 ms]
2.09/1.32	(2) BOUNDS(1, n^2)
2.09/1.32	
2.09/1.32	
2.09/1.32	----------------------------------------
2.09/1.32	
2.09/1.32	(0)
2.09/1.32	Obligation:
2.09/1.32	Complexity Int TRS consisting of the following rules:
2.09/1.32	eval_ax_start(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb0_in(v_.0, v_.01, v_i, v_j, v_n)) :|: TRUE
2.09/1.32	eval_ax_bb0_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb1_in(0, v_.01, v_i, v_j, v_n)) :|: TRUE
2.09/1.32	eval_ax_bb1_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb2_in(v_.0, 0, v_i, v_j, v_n)) :|: TRUE
2.09/1.32	eval_ax_bb2_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb3_in(v_.0, v_.01, v_i, v_j, v_n)) :|: v_.01 < v_n - 1
2.09/1.32	eval_ax_bb2_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb4_in(v_.0, v_.01, v_i, v_j, v_n)) :|: v_.01 >= v_n - 1
2.09/1.32	eval_ax_bb3_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb2_in(v_.0, v_.01 + 1, v_i, v_j, v_n)) :|: TRUE
2.09/1.32	eval_ax_bb4_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb1_in(v_.0 + 1, v_.01, v_i, v_j, v_n)) :|: v_.01 >= v_n - 1 && v_.0 + 1 < v_n - 1
2.09/1.32	eval_ax_bb4_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_.critedge_in(v_.0, v_.01, v_i, v_j, v_n)) :|: v_.01 < v_n - 1
2.09/1.32	eval_ax_bb4_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_.critedge_in(v_.0, v_.01, v_i, v_j, v_n)) :|: v_.0 + 1 >= v_n - 1
2.09/1.32	eval_ax_.critedge_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_stop(v_.0, v_.01, v_i, v_j, v_n)) :|: TRUE
2.09/1.32	
2.09/1.32	The start-symbols are:[eval_ax_start_5]
2.09/1.32	
2.09/1.32	
2.09/1.32	----------------------------------------
2.09/1.32	
2.09/1.32	(1) Koat Proof (FINISHED)
2.09/1.32	YES(?, 7*ar_2 + 2*ar_2^2 + 10)
2.09/1.32	
2.09/1.32	
2.09/1.32	
2.09/1.32	Initial complexity problem:
2.09/1.32	
2.09/1.32	1:	T:
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\ ar_2 >= ar_0 + 3 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.09/1.32	
2.09/1.32		start location:	koat_start
2.09/1.32	
2.09/1.32		leaf cost:	0
2.09/1.32	
2.09/1.32	
2.09/1.32	
2.09/1.32	Testing for reachability in the complexity graph removes the following transition from problem 1:
2.09/1.32	
2.09/1.32		evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32	We thus obtain the following problem:
2.09/1.32	
2.09/1.32	2:	T:
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\ ar_2 >= ar_0 + 3 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.09/1.32	
2.09/1.32		start location:	koat_start
2.09/1.32	
2.09/1.32		leaf cost:	0
2.09/1.32	
2.09/1.32	
2.09/1.32	
2.09/1.32	Repeatedly propagating knowledge in problem 2 produces the following problem:
2.09/1.32	
2.09/1.32	3:	T:
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\ ar_2 >= ar_0 + 3 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)    evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)    evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.09/1.32	
2.09/1.32		start location:	koat_start
2.09/1.32	
2.09/1.32		leaf cost:	0
2.09/1.32	
2.09/1.32	
2.09/1.32	
2.09/1.32	A polynomial rank function with
2.09/1.32	
2.09/1.32		Pol(evalaxcritedgein) = 1
2.09/1.32	
2.09/1.32		Pol(evalaxstop) = 0
2.09/1.32	
2.09/1.32		Pol(evalaxbb4in) = 2
2.09/1.32	
2.09/1.32		Pol(evalaxbb1in) = 2
2.09/1.32	
2.09/1.32		Pol(evalaxbb3in) = 2
2.09/1.32	
2.09/1.32		Pol(evalaxbb2in) = 2
2.09/1.32	
2.09/1.32		Pol(evalaxbb0in) = 2
2.09/1.32	
2.09/1.32		Pol(evalaxstart) = 2
2.09/1.32	
2.09/1.32		Pol(koat_start) = 2
2.09/1.32	
2.09/1.32	orients all transitions weakly and the transitions
2.09/1.32	
2.09/1.32		evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32		evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]
2.09/1.32	
2.09/1.32	strictly and produces the following problem:
2.09/1.32	
2.09/1.32	4:	T:
2.09/1.32	
2.09/1.32			(Comp: 2, Cost: 1)    evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 2, Cost: 1)    evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\ ar_2 >= ar_0 + 3 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)    evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)    evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)    evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.09/1.32	
2.09/1.32		start location:	koat_start
2.09/1.32	
2.09/1.32		leaf cost:	0
2.09/1.32	
2.09/1.32	
2.09/1.32	
2.09/1.32	A polynomial rank function with
2.09/1.32	
2.09/1.32		Pol(evalaxcritedgein) = -V_1 + V_3
2.09/1.32	
2.09/1.32		Pol(evalaxstop) = -V_1 + V_3
2.09/1.32	
2.09/1.32		Pol(evalaxbb4in) = -V_1 + V_3
2.09/1.32	
2.09/1.32		Pol(evalaxbb1in) = -V_1 + V_3
2.09/1.32	
2.09/1.32		Pol(evalaxbb3in) = -V_1 + V_3
2.09/1.32	
2.09/1.32		Pol(evalaxbb2in) = -V_1 + V_3
2.09/1.32	
2.09/1.32		Pol(evalaxbb0in) = V_3
2.09/1.32	
2.09/1.32		Pol(evalaxstart) = V_3
2.09/1.32	
2.09/1.32		Pol(koat_start) = V_3
2.09/1.32	
2.09/1.32	orients all transitions weakly and the transition
2.09/1.32	
2.09/1.32		evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\ ar_2 >= ar_0 + 3 ]
2.09/1.32	
2.09/1.32	strictly and produces the following problem:
2.09/1.32	
2.09/1.32	5:	T:
2.09/1.32	
2.09/1.32			(Comp: 2, Cost: 1)       evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 2, Cost: 1)       evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ar_2, Cost: 1)    evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\ ar_2 >= ar_0 + 3 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)       evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)       evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)       evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)       evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)       evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)       evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.09/1.32	
2.09/1.32		start location:	koat_start
2.09/1.32	
2.09/1.32		leaf cost:	0
2.09/1.32	
2.09/1.32	
2.09/1.32	
2.09/1.32	Repeatedly propagating knowledge in problem 5 produces the following problem:
2.09/1.32	
2.09/1.32	6:	T:
2.09/1.32	
2.09/1.32			(Comp: 2, Cost: 1)           evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 2, Cost: 1)           evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ar_2, Cost: 1)        evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\ ar_2 >= ar_0 + 3 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)           evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)           evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)           evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32			(Comp: ar_2 + 1, Cost: 1)    evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)           evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)           evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.09/1.32	
2.09/1.32		start location:	koat_start
2.09/1.32	
2.09/1.32		leaf cost:	0
2.09/1.32	
2.09/1.32	
2.09/1.32	
2.09/1.32	A polynomial rank function with
2.09/1.32	
2.09/1.32		Pol(evalaxbb3in) = 1
2.09/1.32	
2.09/1.32		Pol(evalaxbb2in) = 1
2.09/1.32	
2.09/1.32		Pol(evalaxbb4in) = 0
2.09/1.32	
2.09/1.32	and size complexities
2.09/1.32	
2.09/1.32		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
2.09/1.32	
2.09/1.32		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
2.09/1.32	
2.09/1.32		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0
2.09/1.32	
2.09/1.32		S("evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.09/1.32	
2.09/1.32		S("evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))", 0-0) = 0
2.09/1.32	
2.09/1.32		S("evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))", 0-1) = ar_1
2.09/1.32	
2.09/1.32		S("evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))", 0-1) = 0
2.09/1.32	
2.09/1.32		S("evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]", 0-1) = ?
2.09/1.32	
2.09/1.32		S("evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]", 0-1) = ?
2.09/1.32	
2.09/1.32		S("evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))", 0-1) = ?
2.09/1.32	
2.09/1.32		S("evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\\ ar_2 >= ar_0 + 3 ]", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\\ ar_2 >= ar_0 + 3 ]", 0-1) = ?
2.09/1.32	
2.09/1.32		S("evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\\ ar_2 >= ar_0 + 3 ]", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]", 0-1) = ?
2.09/1.32	
2.09/1.32		S("evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))", 0-1) = ?
2.09/1.32	
2.09/1.32		S("evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))", 0-2) = ar_2
2.09/1.32	
2.09/1.32	orients the transitions
2.09/1.32	
2.09/1.32		evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))
2.09/1.32	
2.09/1.32		evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]
2.09/1.32	
2.09/1.32		evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32	weakly and the transition
2.09/1.32	
2.09/1.32		evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]
2.09/1.32	
2.09/1.32	strictly and produces the following problem:
2.09/1.32	
2.09/1.32	7:	T:
2.09/1.32	
2.09/1.32			(Comp: 2, Cost: 1)           evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 2, Cost: 1)           evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ar_2, Cost: 1)        evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\ ar_2 >= ar_0 + 3 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)           evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ar_2 + 1, Cost: 1)    evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)           evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32			(Comp: ar_2 + 1, Cost: 1)    evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)           evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)           evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.09/1.32	
2.09/1.32		start location:	koat_start
2.09/1.32	
2.09/1.32		leaf cost:	0
2.09/1.32	
2.09/1.32	
2.09/1.32	
2.09/1.32	A polynomial rank function with
2.09/1.32	
2.09/1.32		Pol(evalaxbb3in) = -V_2 + V_3
2.09/1.32	
2.09/1.32		Pol(evalaxbb2in) = -V_2 + V_3 + 1
2.09/1.32	
2.09/1.32	and size complexities
2.09/1.32	
2.09/1.32		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
2.09/1.32	
2.09/1.32		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
2.09/1.32	
2.09/1.32		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0
2.09/1.32	
2.09/1.32		S("evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.09/1.32	
2.09/1.32		S("evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))", 0-0) = 0
2.09/1.32	
2.09/1.32		S("evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))", 0-1) = ar_1
2.09/1.32	
2.09/1.32		S("evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))", 0-1) = 0
2.09/1.32	
2.09/1.32		S("evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]", 0-1) = ?
2.09/1.32	
2.09/1.32		S("evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]", 0-1) = ?
2.09/1.32	
2.09/1.32		S("evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))", 0-1) = ?
2.09/1.32	
2.09/1.32		S("evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\\ ar_2 >= ar_0 + 3 ]", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\\ ar_2 >= ar_0 + 3 ]", 0-1) = ?
2.09/1.32	
2.09/1.32		S("evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\\ ar_2 >= ar_0 + 3 ]", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]", 0-1) = ?
2.09/1.32	
2.09/1.32		S("evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]", 0-2) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))", 0-0) = ar_2
2.09/1.32	
2.09/1.32		S("evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))", 0-1) = ?
2.09/1.32	
2.09/1.32		S("evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))", 0-2) = ar_2
2.09/1.32	
2.09/1.32	orients the transitions
2.09/1.32	
2.09/1.32		evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))
2.09/1.32	
2.09/1.32		evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32	weakly and the transition
2.09/1.32	
2.09/1.32		evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32	strictly and produces the following problem:
2.09/1.32	
2.09/1.32	8:	T:
2.09/1.32	
2.09/1.32			(Comp: 2, Cost: 1)                      evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 2, Cost: 1)                      evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ar_2, Cost: 1)                   evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\ ar_2 >= ar_0 + 3 ]
2.09/1.32	
2.09/1.32			(Comp: ?, Cost: 1)                      evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ar_2 + 1, Cost: 1)               evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ar_2^2 + 2*ar_2 + 1, Cost: 1)    evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32			(Comp: ar_2 + 1, Cost: 1)               evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)                      evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)                      evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 0)                      koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.09/1.32	
2.09/1.32		start location:	koat_start
2.09/1.32	
2.09/1.32		leaf cost:	0
2.09/1.32	
2.09/1.32	
2.09/1.32	
2.09/1.32	Repeatedly propagating knowledge in problem 8 produces the following problem:
2.09/1.32	
2.09/1.32	9:	T:
2.09/1.32	
2.09/1.32			(Comp: 2, Cost: 1)                      evalaxcritedgein(ar_0, ar_1, ar_2) -> Com_1(evalaxstop(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 2, Cost: 1)                      evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxcritedgein(ar_0, ar_1, ar_2)) [ ar_0 + 2 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ar_2, Cost: 1)                   evalaxbb4in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(ar_0 + 1, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 /\ ar_2 >= ar_0 + 3 ]
2.09/1.32	
2.09/1.32			(Comp: ar_2^2 + 2*ar_2 + 1, Cost: 1)    evalaxbb3in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, ar_1 + 1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: ar_2 + 1, Cost: 1)               evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb4in(ar_0, ar_1, ar_2)) [ ar_1 + 1 >= ar_2 ]
2.09/1.32	
2.09/1.32			(Comp: ar_2^2 + 2*ar_2 + 1, Cost: 1)    evalaxbb2in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb3in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 2 ]
2.09/1.32	
2.09/1.32			(Comp: ar_2 + 1, Cost: 1)               evalaxbb1in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb2in(ar_0, 0, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)                      evalaxbb0in(ar_0, ar_1, ar_2) -> Com_1(evalaxbb1in(0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 1)                      evalaxstart(ar_0, ar_1, ar_2) -> Com_1(evalaxbb0in(ar_0, ar_1, ar_2))
2.09/1.32	
2.09/1.32			(Comp: 1, Cost: 0)                      koat_start(ar_0, ar_1, ar_2) -> Com_1(evalaxstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.09/1.32	
2.09/1.32		start location:	koat_start
2.09/1.32	
2.09/1.32		leaf cost:	0
2.09/1.32	
2.09/1.32	
2.09/1.32	
2.09/1.32	Complexity upper bound 7*ar_2 + 2*ar_2^2 + 10
2.09/1.32	
2.09/1.32	
2.09/1.32	
2.09/1.32	Time: 0.091 sec (SMT: 0.078 sec)
2.09/1.32	
2.09/1.32	
2.09/1.32	----------------------------------------
2.09/1.32	
2.09/1.32	(2)
2.09/1.32	BOUNDS(1, n^2)
2.09/1.33	EOF