2.08/1.26	WORST_CASE(?, O(n^1))
2.08/1.27	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.08/1.27	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.08/1.27	
2.08/1.27	
2.08/1.27	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.08/1.27	
2.08/1.27	(0) CpxIntTrs
2.08/1.27	(1) Koat Proof [FINISHED, 74 ms]
2.08/1.27	(2) BOUNDS(1, n^1)
2.08/1.27	
2.08/1.27	
2.08/1.27	----------------------------------------
2.08/1.27	
2.08/1.27	(0)
2.08/1.27	Obligation:
2.08/1.27	Complexity Int TRS consisting of the following rules:
2.08/1.27	eval_foo_start(v_.0, v_.01, v_oldx, v_x, v_y) -> Com_1(eval_foo_bb0_in(v_.0, v_.01, v_oldx, v_x, v_y)) :|: TRUE
2.08/1.27	eval_foo_bb0_in(v_.0, v_.01, v_oldx, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_x, v_y, v_oldx, v_x, v_y)) :|: TRUE
2.08/1.27	eval_foo_bb1_in(v_.0, v_.01, v_oldx, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.0, v_.01, v_oldx, v_x, v_y)) :|: v_.0 >= 0 && v_.01 >= 0
2.08/1.27	eval_foo_bb1_in(v_.0, v_.01, v_oldx, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_.01, v_oldx, v_x, v_y)) :|: v_.0 < 0
2.08/1.27	eval_foo_bb1_in(v_.0, v_.01, v_oldx, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_.01, v_oldx, v_x, v_y)) :|: v_.01 < 0
2.08/1.27	eval_foo_bb2_in(v_.0, v_.01, v_oldx, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_.01 - 1, v_.0 - 1, v_oldx, v_x, v_y)) :|: TRUE
2.08/1.27	eval_foo_bb3_in(v_.0, v_.01, v_oldx, v_x, v_y) -> Com_1(eval_foo_stop(v_.0, v_.01, v_oldx, v_x, v_y)) :|: TRUE
2.08/1.27	
2.08/1.27	The start-symbols are:[eval_foo_start_5]
2.08/1.27	
2.08/1.27	
2.08/1.27	----------------------------------------
2.08/1.27	
2.08/1.27	(1) Koat Proof (FINISHED)
2.08/1.27	YES(?, 2*ar_1 + 2*ar_3 + 12)
2.08/1.27	
2.08/1.27	
2.08/1.27	
2.08/1.27	Initial complexity problem:
2.08/1.27	
2.08/1.27	1:	T:
2.08/1.27	
2.08/1.27			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.08/1.27	
2.08/1.27			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3))
2.08/1.27	
2.08/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 /\ ar_2 >= 0 ]
2.08/1.27	
2.08/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.08/1.27	
2.08/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ]
2.08/1.27	
2.08/1.27			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_2 - 1, ar_1, ar_0 - 1, ar_3))
2.08/1.27	
2.08/1.27			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.08/1.27	
2.08/1.27			(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 ]
2.08/1.27	
2.08/1.27		start location:	koat_start
2.08/1.27	
2.08/1.27		leaf cost:	0
2.08/1.27	
2.08/1.27	
2.08/1.27	
2.08/1.27	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.08/1.27	
2.08/1.27	2:	T:
2.08/1.27	
2.08/1.27			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.08/1.27	
2.08/1.27			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3))
2.08/1.27	
2.08/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 /\ ar_2 >= 0 ]
2.08/1.27	
2.08/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.08/1.27	
2.08/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ]
2.08/1.27	
2.08/1.27			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_2 - 1, ar_1, ar_0 - 1, ar_3))
2.08/1.27	
2.08/1.27			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.08/1.27	
2.08/1.27			(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 ]
2.08/1.27	
2.08/1.27		start location:	koat_start
2.08/1.27	
2.08/1.27		leaf cost:	0
2.08/1.27	
2.08/1.27	
2.08/1.27	
2.08/1.27	A polynomial rank function with
2.08/1.27	
2.08/1.27		Pol(evalfoostart) = 2
2.08/1.27	
2.08/1.27		Pol(evalfoobb0in) = 2
2.08/1.27	
2.08/1.27		Pol(evalfoobb1in) = 2
2.08/1.27	
2.08/1.27		Pol(evalfoobb2in) = 2
2.08/1.27	
2.08/1.27		Pol(evalfoobb3in) = 1
2.08/1.27	
2.08/1.27		Pol(evalfoostop) = 0
2.08/1.27	
2.08/1.27		Pol(koat_start) = 2
2.08/1.27	
2.08/1.27	orients all transitions weakly and the transitions
2.08/1.27	
2.08/1.27		evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.08/1.27	
2.08/1.27		evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ]
2.08/1.27	
2.08/1.27		evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.08/1.27	
2.08/1.27	strictly and produces the following problem:
2.08/1.28	
2.08/1.28	3:	T:
2.08/1.28	
2.08/1.28			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.08/1.28	
2.08/1.28			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3))
2.08/1.28	
2.08/1.28			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 /\ ar_2 >= 0 ]
2.08/1.28	
2.08/1.28			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.08/1.28	
2.08/1.28			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ]
2.08/1.28	
2.08/1.28			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_2 - 1, ar_1, ar_0 - 1, ar_3))
2.08/1.28	
2.08/1.28			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.08/1.28	
2.08/1.28			(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 ]
2.08/1.28	
2.08/1.28		start location:	koat_start
2.08/1.28	
2.08/1.28		leaf cost:	0
2.08/1.28	
2.08/1.28	
2.08/1.28	
2.08/1.28	A polynomial rank function with
2.08/1.28	
2.08/1.28		Pol(evalfoostart) = V_2 + V_4 + 2
2.08/1.28	
2.08/1.28		Pol(evalfoobb0in) = V_2 + V_4 + 2
2.08/1.28	
2.08/1.28		Pol(evalfoobb1in) = V_1 + V_3 + 2
2.08/1.28	
2.08/1.28		Pol(evalfoobb2in) = V_1 + V_3
2.08/1.28	
2.08/1.28		Pol(evalfoobb3in) = V_1 + V_3
2.08/1.28	
2.08/1.28		Pol(evalfoostop) = V_1 + V_3
2.08/1.28	
2.08/1.28		Pol(koat_start) = V_2 + V_4 + 2
2.08/1.28	
2.08/1.28	orients all transitions weakly and the transition
2.08/1.28	
2.08/1.28		evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 /\ ar_2 >= 0 ]
2.08/1.28	
2.08/1.28	strictly and produces the following problem:
2.08/1.28	
2.08/1.28	4:	T:
2.08/1.28	
2.08/1.28			(Comp: 1, Cost: 1)                  evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.08/1.28	
2.08/1.28			(Comp: 1, Cost: 1)                  evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3))
2.08/1.28	
2.08/1.28			(Comp: ar_1 + ar_3 + 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 /\ ar_2 >= 0 ]
2.08/1.28	
2.08/1.28			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.08/1.28	
2.08/1.28			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ]
2.08/1.28	
2.08/1.28			(Comp: ?, Cost: 1)                  evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_2 - 1, ar_1, ar_0 - 1, ar_3))
2.08/1.28	
2.08/1.28			(Comp: 2, Cost: 1)                  evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.08/1.28	
2.08/1.28			(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 ]
2.08/1.28	
2.08/1.28		start location:	koat_start
2.08/1.28	
2.08/1.28		leaf cost:	0
2.08/1.28	
2.08/1.28	
2.08/1.28	
2.08/1.28	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.08/1.28	
2.08/1.28	5:	T:
2.08/1.28	
2.08/1.28			(Comp: 1, Cost: 1)                  evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.08/1.28	
2.08/1.28			(Comp: 1, Cost: 1)                  evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3))
2.08/1.28	
2.08/1.28			(Comp: ar_1 + ar_3 + 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 /\ ar_2 >= 0 ]
2.08/1.28	
2.08/1.28			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.08/1.28	
2.08/1.28			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ]
2.08/1.28	
2.08/1.28			(Comp: ar_1 + ar_3 + 2, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_2 - 1, ar_1, ar_0 - 1, ar_3))
2.08/1.28	
2.08/1.28			(Comp: 2, Cost: 1)                  evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.08/1.28	
2.08/1.28			(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 ]
2.08/1.28	
2.08/1.28		start location:	koat_start
2.08/1.28	
2.08/1.28		leaf cost:	0
2.08/1.28	
2.08/1.28	
2.08/1.28	
2.08/1.28	Complexity upper bound 2*ar_1 + 2*ar_3 + 12
2.08/1.28	
2.08/1.28	
2.08/1.28	
2.08/1.28	Time: 0.069 sec (SMT: 0.061 sec)
2.08/1.28	
2.08/1.28	
2.08/1.28	----------------------------------------
2.08/1.28	
2.08/1.28	(2)
2.08/1.28	BOUNDS(1, n^1)
2.08/1.29	EOF