2.35/1.31	WORST_CASE(?, O(n^1))
2.35/1.32	proof of /export/starexec/sandbox2/output/output_files/bench.koat
2.35/1.32	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.35/1.32	
2.35/1.32	
2.35/1.32	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.35/1.32	
2.35/1.32	(0) CpxIntTrs
2.35/1.32	(1) Koat Proof [FINISHED, 77 ms]
2.35/1.32	(2) BOUNDS(1, n^1)
2.35/1.32	
2.35/1.32	
2.35/1.32	----------------------------------------
2.35/1.32	
2.35/1.32	(0)
2.35/1.32	Obligation:
2.35/1.32	Complexity Int TRS consisting of the following rules:
2.35/1.32	eval_foo_start(v_.0, v_.1, v_min, v_res, v_x, v_y) -> Com_1(eval_foo_bb0_in(v_.0, v_.1, v_min, v_res, v_x, v_y)) :|: TRUE
2.35/1.32	eval_foo_bb0_in(v_.0, v_.1, v_min, v_res, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_y, v_x, v_min, v_res, v_x, v_y)) :|: v_x < v_y
2.35/1.32	eval_foo_bb0_in(v_.0, v_.1, v_min, v_res, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_y, v_y, v_min, v_res, v_x, v_y)) :|: v_x >= v_y
2.35/1.32	eval_foo_bb1_in(v_.0, v_.1, v_min, v_res, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.0, v_.1, v_min, v_res, v_x, v_y)) :|: v_.1 >= v_.0 && v_.1 <= v_.0
2.35/1.32	eval_foo_bb1_in(v_.0, v_.1, v_min, v_res, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_.1, v_min, v_res, v_x, v_y)) :|: v_.1 < v_.0
2.35/1.32	eval_foo_bb1_in(v_.0, v_.1, v_min, v_res, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_.1, v_min, v_res, v_x, v_y)) :|: v_.1 > v_.0
2.35/1.32	eval_foo_bb2_in(v_.0, v_.1, v_min, v_res, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_.0 + 1, v_x, v_min, v_res, v_x, v_y)) :|: v_x < v_.0 + 1
2.35/1.32	eval_foo_bb2_in(v_.0, v_.1, v_min, v_res, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_.0 + 1, v_.0 + 1, v_min, v_res, v_x, v_y)) :|: v_x >= v_.0 + 1
2.35/1.32	eval_foo_bb3_in(v_.0, v_.1, v_min, v_res, v_x, v_y) -> Com_1(eval_foo_stop(v_.0, v_.1, v_min, v_res, v_x, v_y)) :|: TRUE
2.35/1.32	
2.35/1.32	The start-symbols are:[eval_foo_start_6]
2.35/1.32	
2.35/1.32	
2.35/1.32	----------------------------------------
2.35/1.32	
2.35/1.32	(1) Koat Proof (FINISHED)
2.35/1.32	YES(?, 3*ar_0 + 3*ar_1 + 12)
2.35/1.32	
2.35/1.32	
2.35/1.32	
2.35/1.32	Initial complexity problem:
2.35/1.32	
2.35/1.32	1:	T:
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_0, ar_0)) [ ar_1 >= ar_0 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 = ar_2 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 + 1, ar_1)) [ ar_2 >= ar_1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 + 1, ar_2 + 1)) [ ar_1 >= ar_2 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.35/1.32	
2.35/1.32		start location:	koat_start
2.35/1.32	
2.35/1.32		leaf cost:	0
2.35/1.32	
2.35/1.32	
2.35/1.32	
2.35/1.32	Testing for reachability in the complexity graph removes the following transition from problem 1:
2.35/1.32	
2.35/1.32		evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 + 1 ]
2.35/1.32	
2.35/1.32	We thus obtain the following problem:
2.35/1.32	
2.35/1.32	2:	T:
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 + 1, ar_2 + 1)) [ ar_1 >= ar_2 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 + 1, ar_1)) [ ar_2 >= ar_1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 = ar_2 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_0, ar_0)) [ ar_1 >= ar_0 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.35/1.32	
2.35/1.32		start location:	koat_start
2.35/1.32	
2.35/1.32		leaf cost:	0
2.35/1.32	
2.35/1.32	
2.35/1.32	
2.35/1.32	Repeatedly propagating knowledge in problem 2 produces the following problem:
2.35/1.32	
2.35/1.32	3:	T:
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 + 1, ar_2 + 1)) [ ar_1 >= ar_2 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 + 1, ar_1)) [ ar_2 >= ar_1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 = ar_2 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_0, ar_0)) [ ar_1 >= ar_0 ]
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.35/1.32	
2.35/1.32		start location:	koat_start
2.35/1.32	
2.35/1.32		leaf cost:	0
2.35/1.32	
2.35/1.32	
2.35/1.32	
2.35/1.32	A polynomial rank function with
2.35/1.32	
2.35/1.32		Pol(evalfoobb3in) = 1
2.35/1.32	
2.35/1.32		Pol(evalfoostop) = 0
2.35/1.32	
2.35/1.32		Pol(evalfoobb2in) = 2
2.35/1.32	
2.35/1.32		Pol(evalfoobb1in) = 2
2.35/1.32	
2.35/1.32		Pol(evalfoobb0in) = 2
2.35/1.32	
2.35/1.32		Pol(evalfoostart) = 2
2.35/1.32	
2.35/1.32		Pol(koat_start) = 2
2.35/1.32	
2.35/1.32	orients all transitions weakly and the transitions
2.35/1.32	
2.35/1.32		evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.35/1.32	
2.35/1.32		evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
2.35/1.32	
2.35/1.32	strictly and produces the following problem:
2.35/1.32	
2.35/1.32	4:	T:
2.35/1.32	
2.35/1.32			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 + 1, ar_2 + 1)) [ ar_1 >= ar_2 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 + 1, ar_1)) [ ar_2 >= ar_1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 = ar_2 ]
2.35/1.32	
2.35/1.32			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_0, ar_0)) [ ar_1 >= ar_0 ]
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.35/1.32	
2.35/1.32		start location:	koat_start
2.35/1.32	
2.35/1.32		leaf cost:	0
2.35/1.32	
2.35/1.32	
2.35/1.32	
2.35/1.32	A polynomial rank function with
2.35/1.32	
2.35/1.32		Pol(evalfoobb3in) = V_2 - V_3
2.35/1.32	
2.35/1.32		Pol(evalfoostop) = V_2 - V_3
2.35/1.32	
2.35/1.32		Pol(evalfoobb2in) = V_2 - V_3 + 1
2.35/1.32	
2.35/1.32		Pol(evalfoobb1in) = V_2 - V_3 + 1
2.35/1.32	
2.35/1.32		Pol(evalfoobb0in) = -V_1 + V_2 + 1
2.35/1.32	
2.35/1.32		Pol(evalfoostart) = -V_1 + V_2 + 1
2.35/1.32	
2.35/1.32		Pol(koat_start) = -V_1 + V_2 + 1
2.35/1.32	
2.35/1.32	orients all transitions weakly and the transition
2.35/1.32	
2.35/1.32		evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 + 1, ar_2 + 1)) [ ar_1 >= ar_2 + 1 ]
2.35/1.32	
2.35/1.32	strictly and produces the following problem:
2.35/1.32	
2.35/1.32	5:	T:
2.35/1.32	
2.35/1.32			(Comp: 2, Cost: 1)                  evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.35/1.32	
2.35/1.32			(Comp: ar_0 + ar_1 + 1, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 + 1, ar_2 + 1)) [ ar_1 >= ar_2 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)                  evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 + 1, ar_1)) [ ar_2 >= ar_1 ]
2.35/1.32	
2.35/1.32			(Comp: ?, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 = ar_2 ]
2.35/1.32	
2.35/1.32			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 1)                  evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_0, ar_0)) [ ar_1 >= ar_0 ]
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 1)                  evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 1)                  evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.35/1.32	
2.35/1.32		start location:	koat_start
2.35/1.32	
2.35/1.32		leaf cost:	0
2.35/1.32	
2.35/1.32	
2.35/1.32	
2.35/1.32	Repeatedly propagating knowledge in problem 5 produces the following problem:
2.35/1.32	
2.35/1.32	6:	T:
2.35/1.32	
2.35/1.32			(Comp: 2, Cost: 1)                  evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.35/1.32	
2.35/1.32			(Comp: ar_0 + ar_1 + 1, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 + 1, ar_2 + 1)) [ ar_1 >= ar_2 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: ar_0 + ar_1 + 2, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 + 1, ar_1)) [ ar_2 >= ar_1 ]
2.35/1.32	
2.35/1.32			(Comp: ar_0 + ar_1 + 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 = ar_2 ]
2.35/1.32	
2.35/1.32			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 1)                  evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_0, ar_0)) [ ar_1 >= ar_0 ]
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 1)                  evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 1)                  evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.35/1.32	
2.35/1.32			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.35/1.32	
2.35/1.32		start location:	koat_start
2.35/1.32	
2.35/1.32		leaf cost:	0
2.35/1.32	
2.35/1.32	
2.35/1.32	
2.35/1.32	Complexity upper bound 3*ar_0 + 3*ar_1 + 12
2.35/1.32	
2.35/1.32	
2.35/1.32	
2.35/1.32	Time: 0.093 sec (SMT: 0.082 sec)
2.35/1.32	
2.35/1.32	
2.35/1.32	----------------------------------------
2.35/1.32	
2.35/1.32	(2)
2.35/1.32	BOUNDS(1, n^1)
2.35/1.33	EOF