2.15/1.28	WORST_CASE(?, O(n^1))
2.15/1.29	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.15/1.29	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.15/1.29	
2.15/1.29	
2.15/1.29	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.15/1.29	
2.15/1.29	(0) CpxIntTrs
2.15/1.29	(1) Koat Proof [FINISHED, 67 ms]
2.15/1.29	(2) BOUNDS(1, n^1)
2.15/1.29	
2.15/1.29	
2.15/1.29	----------------------------------------
2.15/1.29	
2.15/1.29	(0)
2.15/1.29	Obligation:
2.15/1.29	Complexity Int TRS consisting of the following rules:
2.15/1.29	eval_t47_start(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb0_in(v_.0, v_flag.0, v_n)) :|: TRUE
2.15/1.29	eval_t47_bb0_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb1_in(v_n, 1, v_n)) :|: TRUE
2.15/1.29	eval_t47_bb1_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb2_in(v_.0, v_flag.0, v_n)) :|: v_flag.0 > 0
2.15/1.29	eval_t47_bb1_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb3_in(v_.0, v_flag.0, v_n)) :|: v_flag.0 <= 0
2.15/1.29	eval_t47_bb2_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb1_in(v_.0 - 1, 1, v_n)) :|: v_.0 > 0
2.15/1.29	eval_t47_bb2_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb1_in(v_.0, 1, v_n)) :|: v_.0 > 0 && v_.0 <= 0
2.15/1.29	eval_t47_bb2_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb1_in(v_.0 - 1, 0, v_n)) :|: v_.0 <= 0 && v_.0 > 0
2.15/1.29	eval_t47_bb2_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb1_in(v_.0, 0, v_n)) :|: v_.0 <= 0
2.15/1.29	eval_t47_bb3_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_stop(v_.0, v_flag.0, v_n)) :|: TRUE
2.15/1.29	
2.15/1.29	The start-symbols are:[eval_t47_start_3]
2.15/1.29	
2.15/1.29	
2.15/1.29	----------------------------------------
2.15/1.29	
2.15/1.29	(1) Koat Proof (FINISHED)
2.15/1.29	YES(?, 2*ar_1 + 8)
2.15/1.29	
2.15/1.29	
2.15/1.29	
2.15/1.29	Initial complexity problem:
2.15/1.29	
2.15/1.29	1:	T:
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1))
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 1)) [ ar_0 >= 1 /\ 0 >= ar_0 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 0)) [ 0 >= ar_0 /\ ar_0 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.15/1.29	
2.15/1.29		start location:	koat_start
2.15/1.29	
2.15/1.29		leaf cost:	0
2.15/1.29	
2.15/1.29	
2.15/1.29	
2.15/1.29	Testing for reachability in the complexity graph removes the following transitions from problem 1:
2.15/1.29	
2.15/1.29		evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 1)) [ ar_0 >= 1 /\ 0 >= ar_0 ]
2.15/1.29	
2.15/1.29		evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 0)) [ 0 >= ar_0 /\ ar_0 >= 1 ]
2.15/1.29	
2.15/1.29	We thus obtain the following problem:
2.15/1.29	
2.15/1.29	2:	T:
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1))
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.15/1.29	
2.15/1.29		start location:	koat_start
2.15/1.29	
2.15/1.29		leaf cost:	0
2.15/1.29	
2.15/1.29	
2.15/1.29	
2.15/1.29	Repeatedly propagating knowledge in problem 2 produces the following problem:
2.15/1.29	
2.15/1.29	3:	T:
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 1)    evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1))
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 1)    evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.15/1.29	
2.15/1.29		start location:	koat_start
2.15/1.29	
2.15/1.29		leaf cost:	0
2.15/1.29	
2.15/1.29	
2.15/1.29	
2.15/1.29	A polynomial rank function with
2.15/1.29	
2.15/1.29		Pol(evalt47bb3in) = 1
2.15/1.29	
2.15/1.29		Pol(evalt47stop) = 0
2.15/1.29	
2.15/1.29		Pol(evalt47bb1in) = 2
2.15/1.29	
2.15/1.29		Pol(evalt47bb2in) = 2
2.15/1.29	
2.15/1.29		Pol(evalt47bb0in) = 2
2.15/1.29	
2.15/1.29		Pol(evalt47start) = 2
2.15/1.29	
2.15/1.29		Pol(koat_start) = 2
2.15/1.29	
2.15/1.29	orients all transitions weakly and the transitions
2.15/1.29	
2.15/1.29		evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29		evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.15/1.29	
2.15/1.29	strictly and produces the following problem:
2.15/1.29	
2.15/1.29	4:	T:
2.15/1.29	
2.15/1.29			(Comp: 2, Cost: 1)    evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: 2, Cost: 1)    evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 1)    evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1))
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 1)    evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.15/1.29	
2.15/1.29		start location:	koat_start
2.15/1.29	
2.15/1.29		leaf cost:	0
2.15/1.29	
2.15/1.29	
2.15/1.29	
2.15/1.29	A polynomial rank function with
2.15/1.29	
2.15/1.29		Pol(evalt47bb3in) = V_3
2.15/1.29	
2.15/1.29		Pol(evalt47stop) = V_3
2.15/1.29	
2.15/1.29		Pol(evalt47bb1in) = V_3
2.15/1.29	
2.15/1.29		Pol(evalt47bb2in) = 1
2.15/1.29	
2.15/1.29		Pol(evalt47bb0in) = 1
2.15/1.29	
2.15/1.29		Pol(evalt47start) = 1
2.15/1.29	
2.15/1.29		Pol(koat_start) = 1
2.15/1.29	
2.15/1.29	orients all transitions weakly and the transition
2.15/1.29	
2.15/1.29		evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
2.15/1.29	
2.15/1.29	strictly and produces the following problem:
2.15/1.29	
2.15/1.29	5:	T:
2.15/1.29	
2.15/1.29			(Comp: 2, Cost: 1)    evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: 2, Cost: 1)    evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 1)    evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)    evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 1)    evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1))
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 1)    evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.15/1.29	
2.15/1.29		start location:	koat_start
2.15/1.29	
2.15/1.29		leaf cost:	0
2.15/1.29	
2.15/1.29	
2.15/1.29	
2.15/1.29	A polynomial rank function with
2.15/1.29	
2.15/1.29		Pol(evalt47bb3in) = V_1
2.15/1.29	
2.15/1.29		Pol(evalt47stop) = V_1
2.15/1.29	
2.15/1.29		Pol(evalt47bb1in) = V_1
2.15/1.29	
2.15/1.29		Pol(evalt47bb2in) = V_1
2.15/1.29	
2.15/1.29		Pol(evalt47bb0in) = V_2
2.15/1.29	
2.15/1.29		Pol(evalt47start) = V_2
2.15/1.29	
2.15/1.29		Pol(koat_start) = V_2
2.15/1.29	
2.15/1.29	orients all transitions weakly and the transition
2.15/1.29	
2.15/1.29		evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
2.15/1.29	
2.15/1.29	strictly and produces the following problem:
2.15/1.29	
2.15/1.29	6:	T:
2.15/1.29	
2.15/1.29			(Comp: 2, Cost: 1)       evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: 2, Cost: 1)       evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 1)       evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
2.15/1.29	
2.15/1.29			(Comp: ar_1, Cost: 1)    evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: ?, Cost: 1)       evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 1)       evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1))
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 1)       evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.15/1.29	
2.15/1.29		start location:	koat_start
2.15/1.29	
2.15/1.29		leaf cost:	0
2.15/1.29	
2.15/1.29	
2.15/1.29	
2.15/1.29	Repeatedly propagating knowledge in problem 6 produces the following problem:
2.15/1.29	
2.15/1.29	7:	T:
2.15/1.29	
2.15/1.29			(Comp: 2, Cost: 1)           evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: 2, Cost: 1)           evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 1)           evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
2.15/1.29	
2.15/1.29			(Comp: ar_1, Cost: 1)        evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: ar_1 + 1, Cost: 1)    evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 1)           evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1))
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 1)           evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2))
2.15/1.29	
2.15/1.29			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.15/1.29	
2.15/1.29		start location:	koat_start
2.15/1.29	
2.15/1.29		leaf cost:	0
2.15/1.29	
2.15/1.29	
2.15/1.29	
2.15/1.29	Complexity upper bound 2*ar_1 + 8
2.15/1.29	
2.15/1.29	
2.15/1.29	
2.15/1.29	Time: 0.083 sec (SMT: 0.076 sec)
2.15/1.29	
2.15/1.29	
2.15/1.29	----------------------------------------
2.15/1.29	
2.15/1.29	(2)
2.15/1.29	BOUNDS(1, n^1)
2.28/1.32	EOF