2.01/1.77	WORST_CASE(?, O(n^1))
2.01/1.78	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.01/1.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.01/1.78	
2.01/1.78	
2.01/1.78	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.01/1.78	
2.01/1.78	(0) CpxIntTrs
2.01/1.78	(1) Koat Proof [FINISHED, 71 ms]
2.01/1.78	(2) BOUNDS(1, n^1)
2.01/1.78	
2.01/1.78	
2.01/1.78	----------------------------------------
2.01/1.78	
2.01/1.78	(0)
2.01/1.78	Obligation:
2.01/1.78	Complexity Int TRS consisting of the following rules:
2.01/1.78	eval_easy2_start(v_.0, v_z) -> Com_1(eval_easy2_bb0_in(v_.0, v_z)) :|: TRUE
2.01/1.78	eval_easy2_bb0_in(v_.0, v_z) -> Com_1(eval_easy2_bb1_in(v_z, v_z)) :|: TRUE
2.01/1.78	eval_easy2_bb1_in(v_.0, v_z) -> Com_1(eval_easy2_bb2_in(v_.0, v_z)) :|: v_.0 > 0
2.01/1.78	eval_easy2_bb1_in(v_.0, v_z) -> Com_1(eval_easy2_bb3_in(v_.0, v_z)) :|: v_.0 <= 0
2.01/1.78	eval_easy2_bb2_in(v_.0, v_z) -> Com_1(eval_easy2_bb1_in(v_.0 - 1, v_z)) :|: TRUE
2.01/1.78	eval_easy2_bb3_in(v_.0, v_z) -> Com_1(eval_easy2_stop(v_.0, v_z)) :|: TRUE
2.01/1.78	
2.01/1.78	The start-symbols are:[eval_easy2_start_2]
2.01/1.78	
2.01/1.78	
2.01/1.78	----------------------------------------
2.01/1.78	
2.01/1.78	(1) Koat Proof (FINISHED)
2.01/1.78	YES(?, 2*ar_1 + 8)
2.01/1.78	
2.01/1.78	
2.01/1.78	
2.01/1.78	Initial complexity problem:
2.01/1.78	
2.01/1.78	1:	T:
2.01/1.78	
2.01/1.78			(Comp: ?, Cost: 1)    evaleasy2start(ar_0, ar_1) -> Com_1(evaleasy2bb0in(ar_0, ar_1))
2.01/1.78	
2.01/1.78			(Comp: ?, Cost: 1)    evaleasy2bb0in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_1, ar_1))
2.01/1.78	
2.01/1.78			(Comp: ?, Cost: 1)    evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb2in(ar_0, ar_1)) [ ar_0 >= 1 ]
2.01/1.78	
2.01/1.78			(Comp: ?, Cost: 1)    evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.01/1.78	
2.01/1.78			(Comp: ?, Cost: 1)    evaleasy2bb2in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_0 - 1, ar_1))
2.01/1.78	
2.01/1.78			(Comp: ?, Cost: 1)    evaleasy2bb3in(ar_0, ar_1) -> Com_1(evaleasy2stop(ar_0, ar_1))
2.01/1.78	
2.01/1.78			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evaleasy2start(ar_0, ar_1)) [ 0 <= 0 ]
2.01/1.78	
2.01/1.78		start location:	koat_start
2.01/1.78	
2.01/1.78		leaf cost:	0
2.01/1.78	
2.01/1.78	
2.01/1.78	
2.01/1.78	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.01/1.78	
2.01/1.78	2:	T:
2.01/1.78	
2.01/1.78			(Comp: 1, Cost: 1)    evaleasy2start(ar_0, ar_1) -> Com_1(evaleasy2bb0in(ar_0, ar_1))
2.01/1.78	
2.01/1.78			(Comp: 1, Cost: 1)    evaleasy2bb0in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_1, ar_1))
2.01/1.78	
2.01/1.78			(Comp: ?, Cost: 1)    evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb2in(ar_0, ar_1)) [ ar_0 >= 1 ]
2.01/1.78	
2.01/1.78			(Comp: ?, Cost: 1)    evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.01/1.78	
2.01/1.78			(Comp: ?, Cost: 1)    evaleasy2bb2in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_0 - 1, ar_1))
2.01/1.78	
2.01/1.78			(Comp: ?, Cost: 1)    evaleasy2bb3in(ar_0, ar_1) -> Com_1(evaleasy2stop(ar_0, ar_1))
2.01/1.78	
2.01/1.78			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evaleasy2start(ar_0, ar_1)) [ 0 <= 0 ]
2.01/1.78	
2.01/1.78		start location:	koat_start
2.01/1.78	
2.01/1.78		leaf cost:	0
2.01/1.78	
2.01/1.78	
2.01/1.78	
2.01/1.78	A polynomial rank function with
2.01/1.78	
2.01/1.78		Pol(evaleasy2start) = 2
2.01/1.78	
2.01/1.78		Pol(evaleasy2bb0in) = 2
2.01/1.78	
2.01/1.78		Pol(evaleasy2bb1in) = 2
2.01/1.78	
2.01/1.78		Pol(evaleasy2bb2in) = 2
2.01/1.78	
2.01/1.78		Pol(evaleasy2bb3in) = 1
2.01/1.78	
2.01/1.78		Pol(evaleasy2stop) = 0
2.01/1.78	
2.01/1.78		Pol(koat_start) = 2
2.01/1.78	
2.01/1.78	orients all transitions weakly and the transitions
2.01/1.78	
2.01/1.78		evaleasy2bb3in(ar_0, ar_1) -> Com_1(evaleasy2stop(ar_0, ar_1))
2.01/1.78	
2.01/1.78		evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.01/1.78	
2.01/1.78	strictly and produces the following problem:
2.01/1.78	
2.01/1.78	3:	T:
2.01/1.78	
2.01/1.78			(Comp: 1, Cost: 1)    evaleasy2start(ar_0, ar_1) -> Com_1(evaleasy2bb0in(ar_0, ar_1))
2.14/1.78	
2.14/1.78			(Comp: 1, Cost: 1)    evaleasy2bb0in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_1, ar_1))
2.14/1.78	
2.14/1.78			(Comp: ?, Cost: 1)    evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb2in(ar_0, ar_1)) [ ar_0 >= 1 ]
2.14/1.78	
2.14/1.78			(Comp: 2, Cost: 1)    evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.14/1.78	
2.14/1.78			(Comp: ?, Cost: 1)    evaleasy2bb2in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_0 - 1, ar_1))
2.14/1.78	
2.14/1.78			(Comp: 2, Cost: 1)    evaleasy2bb3in(ar_0, ar_1) -> Com_1(evaleasy2stop(ar_0, ar_1))
2.14/1.78	
2.14/1.78			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evaleasy2start(ar_0, ar_1)) [ 0 <= 0 ]
2.14/1.78	
2.14/1.78		start location:	koat_start
2.14/1.78	
2.14/1.78		leaf cost:	0
2.14/1.78	
2.14/1.78	
2.14/1.78	
2.14/1.78	A polynomial rank function with
2.14/1.78	
2.14/1.78		Pol(evaleasy2start) = V_2 + 1
2.14/1.78	
2.14/1.78		Pol(evaleasy2bb0in) = V_2 + 1
2.14/1.78	
2.14/1.78		Pol(evaleasy2bb1in) = V_1 + 1
2.14/1.78	
2.14/1.78		Pol(evaleasy2bb2in) = V_1
2.14/1.78	
2.14/1.78		Pol(evaleasy2bb3in) = V_1
2.14/1.78	
2.14/1.78		Pol(evaleasy2stop) = V_1
2.14/1.78	
2.14/1.78		Pol(koat_start) = V_2 + 1
2.14/1.78	
2.14/1.78	orients all transitions weakly and the transition
2.14/1.78	
2.14/1.78		evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb2in(ar_0, ar_1)) [ ar_0 >= 1 ]
2.14/1.79	
2.14/1.79	strictly and produces the following problem:
2.14/1.79	
2.14/1.79	4:	T:
2.14/1.79	
2.14/1.79			(Comp: 1, Cost: 1)           evaleasy2start(ar_0, ar_1) -> Com_1(evaleasy2bb0in(ar_0, ar_1))
2.14/1.79	
2.14/1.79			(Comp: 1, Cost: 1)           evaleasy2bb0in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_1, ar_1))
2.14/1.79	
2.14/1.79			(Comp: ar_1 + 1, Cost: 1)    evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb2in(ar_0, ar_1)) [ ar_0 >= 1 ]
2.14/1.79	
2.14/1.79			(Comp: 2, Cost: 1)           evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.14/1.79	
2.14/1.79			(Comp: ?, Cost: 1)           evaleasy2bb2in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_0 - 1, ar_1))
2.14/1.79	
2.14/1.79			(Comp: 2, Cost: 1)           evaleasy2bb3in(ar_0, ar_1) -> Com_1(evaleasy2stop(ar_0, ar_1))
2.14/1.79	
2.14/1.79			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1) -> Com_1(evaleasy2start(ar_0, ar_1)) [ 0 <= 0 ]
2.14/1.79	
2.14/1.79		start location:	koat_start
2.14/1.79	
2.14/1.79		leaf cost:	0
2.14/1.79	
2.14/1.79	
2.14/1.79	
2.14/1.79	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.14/1.79	
2.14/1.79	5:	T:
2.14/1.79	
2.14/1.79			(Comp: 1, Cost: 1)           evaleasy2start(ar_0, ar_1) -> Com_1(evaleasy2bb0in(ar_0, ar_1))
2.14/1.79	
2.14/1.79			(Comp: 1, Cost: 1)           evaleasy2bb0in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_1, ar_1))
2.14/1.79	
2.14/1.79			(Comp: ar_1 + 1, Cost: 1)    evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb2in(ar_0, ar_1)) [ ar_0 >= 1 ]
2.14/1.79	
2.14/1.79			(Comp: 2, Cost: 1)           evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.14/1.79	
2.14/1.79			(Comp: ar_1 + 1, Cost: 1)    evaleasy2bb2in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_0 - 1, ar_1))
2.14/1.79	
2.14/1.79			(Comp: 2, Cost: 1)           evaleasy2bb3in(ar_0, ar_1) -> Com_1(evaleasy2stop(ar_0, ar_1))
2.14/1.79	
2.14/1.79			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1) -> Com_1(evaleasy2start(ar_0, ar_1)) [ 0 <= 0 ]
2.14/1.79	
2.14/1.79		start location:	koat_start
2.14/1.79	
2.14/1.79		leaf cost:	0
2.14/1.79	
2.14/1.79	
2.14/1.79	
2.14/1.79	Complexity upper bound 2*ar_1 + 8
2.14/1.79	
2.14/1.79	
2.14/1.79	
2.14/1.79	Time: 0.038 sec (SMT: 0.034 sec)
2.14/1.79	
2.14/1.79	
2.14/1.79	----------------------------------------
2.14/1.79	
2.14/1.79	(2)
2.14/1.79	BOUNDS(1, n^1)
2.15/1.81	EOF