1.99/1.36	WORST_CASE(?, O(n^1))
1.99/1.37	proof of /export/starexec/sandbox2/output/output_files/bench.koat
1.99/1.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1.99/1.37	
1.99/1.37	
1.99/1.37	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
1.99/1.37	
1.99/1.37	(0) CpxIntTrs
1.99/1.37	(1) Koat Proof [FINISHED, 71 ms]
1.99/1.37	(2) BOUNDS(1, n^1)
1.99/1.37	
1.99/1.37	
1.99/1.37	----------------------------------------
1.99/1.37	
1.99/1.37	(0)
1.99/1.37	Obligation:
1.99/1.37	Complexity Int TRS consisting of the following rules:
1.99/1.37	eval_foo_start(v_.0, v_x, v_y) -> Com_1(eval_foo_bb0_in(v_.0, v_x, v_y)) :|: TRUE
1.99/1.37	eval_foo_bb0_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_y, v_x, v_y)) :|: TRUE
1.99/1.37	eval_foo_bb1_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.0, v_x, v_y)) :|: v_x > v_.0
1.99/1.37	eval_foo_bb1_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_x, v_y)) :|: v_x <= v_.0
1.99/1.37	eval_foo_bb2_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_.0 + 1, v_x, v_y)) :|: TRUE
1.99/1.37	eval_foo_bb3_in(v_.0, v_x, v_y) -> Com_1(eval_foo_stop(v_.0, v_x, v_y)) :|: TRUE
1.99/1.37	
1.99/1.37	The start-symbols are:[eval_foo_start_3]
1.99/1.37	
1.99/1.37	
1.99/1.37	----------------------------------------
1.99/1.37	
1.99/1.37	(1) Koat Proof (FINISHED)
1.99/1.37	YES(?, 2*ar_1 + 2*ar_2 + 8)
1.99/1.37	
1.99/1.37	
1.99/1.37	
1.99/1.37	Initial complexity problem:
1.99/1.37	
1.99/1.37	1:	T:
1.99/1.37	
1.99/1.37			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
1.99/1.37	
1.99/1.37			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
1.99/1.37	
1.99/1.37			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
1.99/1.37	
1.99/1.37		start location:	koat_start
1.99/1.37	
1.99/1.37		leaf cost:	0
1.99/1.37	
1.99/1.37	
1.99/1.37	
1.99/1.37	Repeatedly propagating knowledge in problem 1 produces the following problem:
1.99/1.37	
1.99/1.37	2:	T:
1.99/1.37	
1.99/1.37			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
1.99/1.37	
1.99/1.37			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
1.99/1.37	
1.99/1.37			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
1.99/1.37	
1.99/1.37		start location:	koat_start
1.99/1.37	
1.99/1.37		leaf cost:	0
1.99/1.37	
1.99/1.37	
1.99/1.37	
1.99/1.37	A polynomial rank function with
1.99/1.37	
1.99/1.37		Pol(evalfoostart) = 2
1.99/1.37	
1.99/1.37		Pol(evalfoobb0in) = 2
1.99/1.37	
1.99/1.37		Pol(evalfoobb1in) = 2
1.99/1.37	
1.99/1.37		Pol(evalfoobb2in) = 2
1.99/1.37	
1.99/1.37		Pol(evalfoobb3in) = 1
1.99/1.37	
1.99/1.37		Pol(evalfoostop) = 0
1.99/1.37	
1.99/1.37		Pol(koat_start) = 2
1.99/1.37	
1.99/1.37	orients all transitions weakly and the transitions
1.99/1.37	
1.99/1.37		evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
1.99/1.37	
1.99/1.37		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
1.99/1.37	
1.99/1.37	strictly and produces the following problem:
1.99/1.37	
1.99/1.37	3:	T:
1.99/1.37	
1.99/1.37			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
1.99/1.37	
1.99/1.37			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
1.99/1.37	
1.99/1.37			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
1.99/1.37	
1.99/1.37		start location:	koat_start
1.99/1.37	
1.99/1.37		leaf cost:	0
1.99/1.37	
1.99/1.37	
1.99/1.37	
1.99/1.37	A polynomial rank function with
1.99/1.37	
1.99/1.37		Pol(evalfoostart) = -V_2 + V_3 + 1
1.99/1.37	
1.99/1.37		Pol(evalfoobb0in) = -V_2 + V_3 + 1
1.99/1.37	
1.99/1.37		Pol(evalfoobb1in) = -V_1 + V_3 + 1
1.99/1.37	
1.99/1.37		Pol(evalfoobb2in) = -V_1 + V_3
1.99/1.37	
1.99/1.37		Pol(evalfoobb3in) = -V_1 + V_3
1.99/1.37	
1.99/1.37		Pol(evalfoostop) = -V_1 + V_3
1.99/1.37	
1.99/1.37		Pol(koat_start) = -V_2 + V_3 + 1
1.99/1.37	
1.99/1.37	orients all transitions weakly and the transition
1.99/1.37	
1.99/1.37		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
1.99/1.37	
1.99/1.37	strictly and produces the following problem:
1.99/1.37	
1.99/1.37	4:	T:
1.99/1.37	
1.99/1.37			(Comp: 1, Cost: 1)                  evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: 1, Cost: 1)                  evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: ar_1 + ar_2 + 1, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
1.99/1.37	
1.99/1.37			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
1.99/1.37	
1.99/1.37			(Comp: ?, Cost: 1)                  evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: 2, Cost: 1)                  evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
1.99/1.37	
1.99/1.37		start location:	koat_start
1.99/1.37	
1.99/1.37		leaf cost:	0
1.99/1.37	
1.99/1.37	
1.99/1.37	
1.99/1.37	Repeatedly propagating knowledge in problem 4 produces the following problem:
1.99/1.37	
1.99/1.37	5:	T:
1.99/1.37	
1.99/1.37			(Comp: 1, Cost: 1)                  evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: 1, Cost: 1)                  evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: ar_1 + ar_2 + 1, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
1.99/1.37	
1.99/1.37			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
1.99/1.37	
1.99/1.37			(Comp: ar_1 + ar_2 + 1, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: 2, Cost: 1)                  evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
1.99/1.37	
1.99/1.37			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
1.99/1.37	
1.99/1.37		start location:	koat_start
1.99/1.37	
1.99/1.37		leaf cost:	0
1.99/1.37	
1.99/1.37	
1.99/1.37	
1.99/1.37	Complexity upper bound 2*ar_1 + 2*ar_2 + 8
1.99/1.37	
1.99/1.37	
1.99/1.37	
1.99/1.37	Time: 0.049 sec (SMT: 0.043 sec)
1.99/1.37	
1.99/1.37	
1.99/1.37	----------------------------------------
1.99/1.37	
1.99/1.37	(2)
1.99/1.37	BOUNDS(1, n^1)
2.03/1.39	EOF