2.34/1.31	WORST_CASE(?, O(n^1))
2.34/1.33	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.34/1.33	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.34/1.33	
2.34/1.33	
2.34/1.33	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.34/1.33	
2.34/1.33	(0) CpxIntTrs
2.34/1.33	(1) Koat Proof [FINISHED, 67 ms]
2.34/1.33	(2) BOUNDS(1, n^1)
2.34/1.33	
2.34/1.33	
2.34/1.33	----------------------------------------
2.34/1.33	
2.34/1.33	(0)
2.34/1.33	Obligation:
2.34/1.33	Complexity Int TRS consisting of the following rules:
2.34/1.33	eval_foo_start(v_.0, v_.02, v_c, v_flag, v_x, v_y) -> Com_1(eval_foo_bb0_in(v_.0, v_.02, v_c, v_flag, v_x, v_y)) :|: TRUE
2.34/1.33	eval_foo_bb0_in(v_.0, v_.02, v_c, v_flag, v_x, v_y) -> Com_1(eval_foo_bb1_in(1, v_x, v_c, v_flag, v_x, v_y)) :|: TRUE
2.34/1.33	eval_foo_bb1_in(v_.0, v_.02, v_c, v_flag, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.0, v_.02, v_c, v_flag, v_x, v_y)) :|: v_.0 < 0
2.34/1.33	eval_foo_bb1_in(v_.0, v_.02, v_c, v_flag, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.0, v_.02, v_c, v_flag, v_x, v_y)) :|: v_.0 > 0
2.34/1.33	eval_foo_bb1_in(v_.0, v_.02, v_c, v_flag, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_.02, v_c, v_flag, v_x, v_y)) :|: v_.0 >= 0 && v_.0 <= 0
2.34/1.33	eval_foo_bb2_in(v_.0, v_.02, v_c, v_flag, v_x, v_y) -> Com_1(eval_foo_bb1_in(0, v_.02 + 1, v_c, v_flag, v_x, v_y)) :|: v_.02 >= v_y
2.34/1.33	eval_foo_bb2_in(v_.0, v_.02, v_c, v_flag, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_.0, v_.02 + 1, v_c, v_flag, v_x, v_y)) :|: v_.02 < v_y
2.34/1.33	eval_foo_bb3_in(v_.0, v_.02, v_c, v_flag, v_x, v_y) -> Com_1(eval_foo_stop(v_.0, v_.02, v_c, v_flag, v_x, v_y)) :|: TRUE
2.34/1.33	
2.34/1.33	The start-symbols are:[eval_foo_start_6]
2.34/1.33	
2.34/1.33	
2.34/1.33	----------------------------------------
2.34/1.33	
2.34/1.33	(1) Koat Proof (FINISHED)
2.34/1.33	YES(?, 5*ar_2 + 5*ar_3 + 8)
2.34/1.33	
2.34/1.33	
2.34/1.33	
2.34/1.33	Initial complexity problem:
2.34/1.33	
2.34/1.33	1:	T:
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(1, ar_2, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.34/1.33	
2.34/1.33			(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 >= 1 ]
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 = 0 ]
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(0, ar_1 + 1, ar_2, ar_3)) [ ar_1 >= ar_3 ]
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ]
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(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 ]
2.34/1.33	
2.34/1.33		start location:	koat_start
2.34/1.33	
2.34/1.33		leaf cost:	0
2.34/1.33	
2.34/1.33	
2.34/1.33	
2.34/1.33	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.34/1.33	
2.34/1.33	2:	T:
2.34/1.33	
2.34/1.33			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(1, ar_2, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.34/1.33	
2.34/1.33			(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 >= 1 ]
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 = 0 ]
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(0, ar_1 + 1, ar_2, ar_3)) [ ar_1 >= ar_3 ]
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ]
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(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 ]
2.34/1.33	
2.34/1.33		start location:	koat_start
2.34/1.33	
2.34/1.33		leaf cost:	0
2.34/1.33	
2.34/1.33	
2.34/1.33	
2.34/1.33	A polynomial rank function with
2.34/1.33	
2.34/1.33		Pol(evalfoostart) = 2
2.34/1.33	
2.34/1.33		Pol(evalfoobb0in) = 2
2.34/1.33	
2.34/1.33		Pol(evalfoobb1in) = 2
2.34/1.33	
2.34/1.33		Pol(evalfoobb2in) = 2
2.34/1.33	
2.34/1.33		Pol(evalfoobb3in) = 1
2.34/1.33	
2.34/1.33		Pol(evalfoostop) = 0
2.34/1.33	
2.34/1.33		Pol(koat_start) = 2
2.34/1.33	
2.34/1.33	orients all transitions weakly and the transitions
2.34/1.33	
2.34/1.33		evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.34/1.33	
2.34/1.33		evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 = 0 ]
2.34/1.33	
2.34/1.33	strictly and produces the following problem:
2.34/1.33	
2.34/1.33	3:	T:
2.34/1.33	
2.34/1.33			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(1, ar_2, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.34/1.33	
2.34/1.33			(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 >= 1 ]
2.34/1.33	
2.34/1.33			(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_0 = 0 ]
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(0, ar_1 + 1, ar_2, ar_3)) [ ar_1 >= ar_3 ]
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ]
2.34/1.33	
2.34/1.33			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(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 ]
2.34/1.33	
2.34/1.33		start location:	koat_start
2.34/1.33	
2.34/1.33		leaf cost:	0
2.34/1.33	
2.34/1.33	
2.34/1.33	
2.34/1.33	A polynomial rank function with
2.34/1.33	
2.34/1.33		Pol(evalfoostart) = -V_3 + V_4
2.34/1.33	
2.34/1.33		Pol(evalfoobb0in) = -V_3 + V_4
2.34/1.33	
2.34/1.33		Pol(evalfoobb1in) = -V_2 + V_4
2.34/1.33	
2.34/1.33		Pol(evalfoobb2in) = -V_2 + V_4
2.34/1.33	
2.34/1.33		Pol(evalfoobb3in) = -V_2 + V_4
2.34/1.33	
2.34/1.33		Pol(evalfoostop) = -V_2 + V_4
2.34/1.33	
2.34/1.33		Pol(koat_start) = -V_3 + V_4
2.34/1.33	
2.34/1.33	orients all transitions weakly and the transition
2.34/1.33	
2.34/1.33		evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ]
2.34/1.33	
2.34/1.33	strictly and produces the following problem:
2.34/1.33	
2.34/1.33	4:	T:
2.34/1.33	
2.34/1.33			(Comp: 1, Cost: 1)              evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(Comp: 1, Cost: 1)              evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(1, ar_2, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)              evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.34/1.33	
2.34/1.33			(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 >= 1 ]
2.34/1.33	
2.34/1.33			(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_0 = 0 ]
2.34/1.33	
2.34/1.33			(Comp: ?, Cost: 1)              evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(0, ar_1 + 1, ar_2, ar_3)) [ ar_1 >= ar_3 ]
2.34/1.33	
2.34/1.33			(Comp: ar_2 + ar_3, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ]
2.34/1.33	
2.34/1.33			(Comp: 2, Cost: 1)              evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(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 ]
2.34/1.33	
2.34/1.33		start location:	koat_start
2.34/1.33	
2.34/1.33		leaf cost:	0
2.34/1.33	
2.34/1.33	
2.34/1.33	
2.34/1.33	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.34/1.33	
2.34/1.33	5:	T:
2.34/1.33	
2.34/1.33			(Comp: 1, Cost: 1)                      evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(Comp: 1, Cost: 1)                      evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(1, ar_2, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(Comp: ar_2 + ar_3, Cost: 1)            evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.34/1.33	
2.34/1.33			(Comp: ar_2 + ar_3 + 1, Cost: 1)        evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 1 ]
2.34/1.33	
2.34/1.33			(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_0 = 0 ]
2.34/1.33	
2.34/1.33			(Comp: 2*ar_2 + 2*ar_3 + 1, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(0, ar_1 + 1, ar_2, ar_3)) [ ar_1 >= ar_3 ]
2.34/1.33	
2.34/1.33			(Comp: ar_2 + ar_3, Cost: 1)            evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ]
2.34/1.33	
2.34/1.33			(Comp: 2, Cost: 1)                      evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.34/1.33	
2.34/1.33			(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 ]
2.34/1.33	
2.34/1.33		start location:	koat_start
2.34/1.33	
2.34/1.33		leaf cost:	0
2.34/1.33	
2.34/1.33	
2.34/1.33	
2.34/1.33	Complexity upper bound 5*ar_2 + 5*ar_3 + 8
2.34/1.33	
2.34/1.33	
2.34/1.33	
2.34/1.33	Time: 0.094 sec (SMT: 0.084 sec)
2.34/1.33	
2.34/1.33	
2.34/1.33	----------------------------------------
2.34/1.33	
2.34/1.33	(2)
2.34/1.33	BOUNDS(1, n^1)
2.34/1.35	EOF