2.26/1.26	WORST_CASE(?, O(n^1))
2.26/1.27	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.26/1.27	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.26/1.27	
2.26/1.27	
2.26/1.27	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.26/1.27	
2.26/1.27	(0) CpxIntTrs
2.26/1.27	(1) Koat Proof [FINISHED, 76 ms]
2.26/1.27	(2) BOUNDS(1, n^1)
2.26/1.27	
2.26/1.27	
2.26/1.27	----------------------------------------
2.26/1.27	
2.26/1.27	(0)
2.26/1.27	Obligation:
2.26/1.27	Complexity Int TRS consisting of the following rules:
2.26/1.27	eval_foo_start(v_.01, v_.02, v_M, v_N, v_a, v_b, v_c, v_i, v_j) -> Com_1(eval_foo_bb0_in(v_.01, v_.02, v_M, v_N, v_a, v_b, v_c, v_i, v_j)) :|: TRUE
2.26/1.27	eval_foo_bb0_in(v_.01, v_.02, v_M, v_N, v_a, v_b, v_c, v_i, v_j) -> Com_1(eval_foo_bb1_in(v_i, v_j, v_M, v_N, v_a, v_b, v_c, v_i, v_j)) :|: TRUE
2.26/1.27	eval_foo_bb1_in(v_.01, v_.02, v_M, v_N, v_a, v_b, v_c, v_i, v_j) -> Com_1(eval_foo_bb2_in(v_.01, v_.02, v_M, v_N, v_a, v_b, v_c, v_i, v_j)) :|: v_.01 < v_M
2.26/1.27	eval_foo_bb1_in(v_.01, v_.02, v_M, v_N, v_a, v_b, v_c, v_i, v_j) -> Com_1(eval_foo_bb2_in(v_.01, v_.02, v_M, v_N, v_a, v_b, v_c, v_i, v_j)) :|: v_.02 < v_N
2.26/1.27	eval_foo_bb1_in(v_.01, v_.02, v_M, v_N, v_a, v_b, v_c, v_i, v_j) -> Com_1(eval_foo_bb3_in(v_.01, v_.02, v_M, v_N, v_a, v_b, v_c, v_i, v_j)) :|: v_.01 >= v_M && v_.02 >= v_N
2.26/1.27	eval_foo_bb2_in(v_.01, v_.02, v_M, v_N, v_a, v_b, v_c, v_i, v_j) -> Com_1(eval_foo_bb1_in(v_.01 + 1, v_.02 + 1, v_M, v_N, v_a, v_b, v_c, v_i, v_j)) :|: TRUE
2.26/1.27	eval_foo_bb3_in(v_.01, v_.02, v_M, v_N, v_a, v_b, v_c, v_i, v_j) -> Com_1(eval_foo_stop(v_.01, v_.02, v_M, v_N, v_a, v_b, v_c, v_i, v_j)) :|: TRUE
2.26/1.27	
2.26/1.27	The start-symbols are:[eval_foo_start_9]
2.26/1.27	
2.26/1.27	
2.26/1.27	----------------------------------------
2.26/1.27	
2.26/1.27	(1) Koat Proof (FINISHED)
2.26/1.27	YES(?, 2*ar_1 + 2*ar_4 + 2*ar_3 + 2*ar_5 + 10)
2.26/1.27	
2.26/1.27	
2.26/1.27	
2.26/1.27	Initial complexity problem:
2.26/1.27	
2.26/1.27	1:	T:
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 >= ar_0 + 1 ]
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 ]
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_4 /\ ar_2 >= ar_5 ]
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.26/1.27	
2.26/1.27		start location:	koat_start
2.26/1.27	
2.26/1.27		leaf cost:	0
2.26/1.27	
2.26/1.27	
2.26/1.27	
2.26/1.27	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.26/1.27	
2.26/1.27	2:	T:
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 >= ar_0 + 1 ]
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 ]
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_4 /\ ar_2 >= ar_5 ]
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.26/1.27	
2.26/1.27		start location:	koat_start
2.26/1.27	
2.26/1.27		leaf cost:	0
2.26/1.27	
2.26/1.27	
2.26/1.27	
2.26/1.27	A polynomial rank function with
2.26/1.27	
2.26/1.27		Pol(evalfoostart) = 2
2.26/1.27	
2.26/1.27		Pol(evalfoobb0in) = 2
2.26/1.27	
2.26/1.27		Pol(evalfoobb1in) = 2
2.26/1.27	
2.26/1.27		Pol(evalfoobb2in) = 2
2.26/1.27	
2.26/1.27		Pol(evalfoobb3in) = 1
2.26/1.27	
2.26/1.27		Pol(evalfoostop) = 0
2.26/1.27	
2.26/1.27		Pol(koat_start) = 2
2.26/1.27	
2.26/1.27	orients all transitions weakly and the transitions
2.26/1.27	
2.26/1.27		evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27		evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_4 /\ ar_2 >= ar_5 ]
2.26/1.27	
2.26/1.27	strictly and produces the following problem:
2.26/1.27	
2.26/1.27	3:	T:
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 >= ar_0 + 1 ]
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 ]
2.26/1.27	
2.26/1.27			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_4 /\ ar_2 >= ar_5 ]
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.26/1.27	
2.26/1.27		start location:	koat_start
2.26/1.27	
2.26/1.27		leaf cost:	0
2.26/1.27	
2.26/1.27	
2.26/1.27	
2.26/1.27	A polynomial rank function with
2.26/1.27	
2.26/1.27		Pol(evalfoostart) = -V_2 + V_5 + 1
2.26/1.27	
2.26/1.27		Pol(evalfoobb0in) = -V_2 + V_5 + 1
2.26/1.27	
2.26/1.27		Pol(evalfoobb1in) = -V_1 + V_5 + 1
2.26/1.27	
2.26/1.27		Pol(evalfoobb2in) = -V_1 + V_5
2.26/1.27	
2.26/1.27		Pol(evalfoobb3in) = -V_1 + V_5
2.26/1.27	
2.26/1.27		Pol(evalfoostop) = -V_1 + V_5
2.26/1.27	
2.26/1.27		Pol(koat_start) = -V_2 + V_5 + 1
2.26/1.27	
2.26/1.27	orients all transitions weakly and the transition
2.26/1.27	
2.26/1.27		evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 >= ar_0 + 1 ]
2.26/1.27	
2.26/1.27	strictly and produces the following problem:
2.26/1.27	
2.26/1.27	4:	T:
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 1)                  evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 1)                  evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: ar_1 + ar_4 + 1, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 >= ar_0 + 1 ]
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 ]
2.26/1.27	
2.26/1.27			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_4 /\ ar_2 >= ar_5 ]
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)                  evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 2, Cost: 1)                  evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.26/1.27	
2.26/1.27		start location:	koat_start
2.26/1.27	
2.26/1.27		leaf cost:	0
2.26/1.27	
2.26/1.27	
2.26/1.27	
2.26/1.27	A polynomial rank function with
2.26/1.27	
2.26/1.27		Pol(evalfoostart) = -V_4 + V_6 + 1
2.26/1.27	
2.26/1.27		Pol(evalfoobb0in) = -V_4 + V_6 + 1
2.26/1.27	
2.26/1.27		Pol(evalfoobb1in) = -V_3 + V_6 + 1
2.26/1.27	
2.26/1.27		Pol(evalfoobb2in) = -V_3 + V_6
2.26/1.27	
2.26/1.27		Pol(evalfoobb3in) = -V_3 + V_6
2.26/1.27	
2.26/1.27		Pol(evalfoostop) = -V_3 + V_6
2.26/1.27	
2.26/1.27		Pol(koat_start) = -V_4 + V_6 + 1
2.26/1.27	
2.26/1.27	orients all transitions weakly and the transition
2.26/1.27	
2.26/1.27		evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 ]
2.26/1.27	
2.26/1.27	strictly and produces the following problem:
2.26/1.27	
2.26/1.27	5:	T:
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 1)                  evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 1)                  evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: ar_1 + ar_4 + 1, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 >= ar_0 + 1 ]
2.26/1.27	
2.26/1.27			(Comp: ar_3 + ar_5 + 1, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 ]
2.26/1.27	
2.26/1.27			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_4 /\ ar_2 >= ar_5 ]
2.26/1.27	
2.26/1.27			(Comp: ?, Cost: 1)                  evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 2, Cost: 1)                  evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.26/1.27	
2.26/1.27		start location:	koat_start
2.26/1.27	
2.26/1.27		leaf cost:	0
2.26/1.27	
2.26/1.27	
2.26/1.27	
2.26/1.27	Repeatedly propagating knowledge in problem 5 produces the following problem:
2.26/1.27	
2.26/1.27	6:	T:
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 1)                                evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 1)                                evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: ar_1 + ar_4 + 1, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 >= ar_0 + 1 ]
2.26/1.27	
2.26/1.27			(Comp: ar_3 + ar_5 + 1, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 ]
2.26/1.27	
2.26/1.27			(Comp: 2, Cost: 1)                                evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_4 /\ ar_2 >= ar_5 ]
2.26/1.27	
2.26/1.27			(Comp: ar_3 + ar_5 + ar_1 + ar_4 + 2, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 2, Cost: 1)                                evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.26/1.27	
2.26/1.27			(Comp: 1, Cost: 0)                                koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.26/1.27	
2.26/1.27		start location:	koat_start
2.26/1.27	
2.26/1.27		leaf cost:	0
2.26/1.27	
2.26/1.27	
2.26/1.27	
2.26/1.27	Complexity upper bound 2*ar_1 + 2*ar_4 + 2*ar_3 + 2*ar_5 + 10
2.26/1.27	
2.26/1.27	
2.26/1.27	
2.26/1.27	Time: 0.108 sec (SMT: 0.090 sec)
2.26/1.27	
2.26/1.27	
2.26/1.27	----------------------------------------
2.26/1.27	
2.26/1.27	(2)
2.26/1.27	BOUNDS(1, n^1)
2.30/1.31	EOF