8.46/6.37	WORST_CASE(?, O(n^2))
8.46/6.38	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
8.46/6.38	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.46/6.38	
8.46/6.38	
8.46/6.38	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, max(1, 3 + 2 * Arg_0) + nat(2 * Arg_0) + nat(Arg_0 * max(2 * Arg_0, -2) + max(2 * Arg_0, -2))).
8.46/6.38	
8.46/6.38	(0) CpxIntTrs
8.46/6.38	(1) Koat2 Proof [FINISHED, 227 ms]
8.46/6.38	(2) BOUNDS(1, max(1, 3 + 2 * Arg_0) + nat(2 * Arg_0) + nat(Arg_0 * max(2 * Arg_0, -2) + max(2 * Arg_0, -2)))
8.46/6.38	
8.46/6.38	
8.46/6.38	----------------------------------------
8.46/6.38	
8.46/6.38	(0)
8.46/6.38	Obligation:
8.46/6.38	Complexity Int TRS consisting of the following rules:
8.46/6.38	eval1(A, B) -> Com_1(eval2(A, 1)) :|: A >= 0
8.46/6.38	eval2(A, B) -> Com_1(eval2(A, 2 * B)) :|: A >= 0 && B >= 1 && A >= B + 1
8.46/6.38	eval2(A, B) -> Com_1(eval1(A - 1, B)) :|: A >= 0 && B >= 1 && B >= A
8.46/6.38	start(A, B) -> Com_1(eval1(A, B)) :|: TRUE
8.46/6.38	
8.46/6.38	The start-symbols are:[start_2]
8.46/6.38	
8.46/6.38	
8.46/6.38	----------------------------------------
8.46/6.38	
8.46/6.38	(1) Koat2 Proof (FINISHED)
8.46/6.38	YES( ?, 1+2*max([0, 1+Arg_0])+max([0, 2*Arg_0])+max([0, (1+Arg_0)*max([-2, 2*Arg_0])]) {O(n^2)})
8.46/6.38	
8.46/6.38	
8.46/6.38	
8.46/6.38	Initial Complexity Problem:
8.46/6.38	
8.46/6.38	  Start:  start  
8.46/6.38	
8.46/6.38	  Program_Vars:  Arg_0, Arg_1  
8.46/6.38	
8.46/6.38	  Temp_Vars:    
8.46/6.38	
8.46/6.38	  Locations:  eval1, eval2, start  
8.46/6.38	
8.46/6.38	  Transitions:
8.46/6.38	
8.46/6.38	  eval1(Arg_0,Arg_1) -> eval2(Arg_0,1):|:0 <= Arg_0
8.46/6.38	
8.46/6.38	  eval2(Arg_0,Arg_1) -> eval1(Arg_0-1,Arg_1):|:1 <= Arg_1 && 1 <= Arg_0+Arg_1 && 0 <= Arg_0 && 0 <= Arg_0 && 1 <= Arg_1 && Arg_0 <= Arg_1
8.46/6.38	
8.46/6.38	  eval2(Arg_0,Arg_1) -> eval2(Arg_0,(2)*Arg_1):|:1 <= Arg_1 && 1 <= Arg_0+Arg_1 && 0 <= Arg_0 && 0 <= Arg_0 && 1 <= Arg_1 && Arg_1+1 <= Arg_0
8.46/6.38	
8.46/6.38	  start(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:  
8.46/6.38	
8.46/6.38	
8.46/6.38	
8.46/6.38	Timebounds: 
8.46/6.38	
8.46/6.38	  Overall timebound: 1+2*max([0, 1+Arg_0])+max([0, 2*Arg_0])+max([0, (1+Arg_0)*max([-2, 2*Arg_0])]) {O(n^2)}
8.46/6.38	
8.46/6.38	  0: eval1->eval2: max([0, 1+Arg_0]) {O(n)}
8.46/6.38	
8.46/6.38	  1: eval2->eval2: max([0, 2*Arg_0])+max([0, (1+Arg_0)*max([-2, 2*Arg_0])]) {O(n^2)}
8.46/6.38	
8.46/6.38	  2: eval2->eval1: max([0, 1+Arg_0]) {O(n)}
8.46/6.38	
8.46/6.38	  3: start->eval1: 1 {O(1)}
8.46/6.38	
8.46/6.38	
8.46/6.38	
8.46/6.38	Costbounds:
8.46/6.38	
8.46/6.38	  Overall costbound: 1+2*max([0, 1+Arg_0])+max([0, 2*Arg_0])+max([0, (1+Arg_0)*max([-2, 2*Arg_0])]) {O(n^2)}
8.46/6.38	
8.46/6.38	  0: eval1->eval2: max([0, 1+Arg_0]) {O(n)}
8.46/6.38	
8.46/6.38	  1: eval2->eval2: max([0, 2*Arg_0])+max([0, (1+Arg_0)*max([-2, 2*Arg_0])]) {O(n^2)}
8.46/6.38	
8.46/6.38	  2: eval2->eval1: max([0, 1+Arg_0]) {O(n)}
8.46/6.38	
8.46/6.38	  3: start->eval1: 1 {O(1)}
8.46/6.38	
8.46/6.38	
8.46/6.38	
8.46/6.38	Sizebounds:
8.46/6.38	
8.46/6.38	`Lower:
8.46/6.38	
8.46/6.38	  0: eval1->eval2, Arg_0: 0 {O(1)}
8.46/6.38	
8.46/6.38	  0: eval1->eval2, Arg_1: 1 {O(1)}
8.46/6.38	
8.46/6.38	  1: eval2->eval2, Arg_0: 2 {O(1)}
8.46/6.38	
8.46/6.38	  1: eval2->eval2, Arg_1: 2 {O(1)}
8.46/6.38	
8.46/6.38	  2: eval2->eval1, Arg_0: -1 {O(1)}
8.46/6.38	
8.46/6.38	  2: eval2->eval1, Arg_1: 1 {O(1)}
8.46/6.38	
8.46/6.38	  3: start->eval1, Arg_0: Arg_0 {O(n)}
8.46/6.38	
8.46/6.38	  3: start->eval1, Arg_1: Arg_1 {O(n)}
8.46/6.38	
8.46/6.38	`Upper:
8.46/6.38	
8.46/6.38	  0: eval1->eval2, Arg_0: Arg_0 {O(n)}
8.46/6.38	
8.46/6.38	  0: eval1->eval2, Arg_1: 1 {O(1)}
8.46/6.38	
8.46/6.38	  1: eval2->eval2, Arg_0: Arg_0 {O(n)}
8.46/6.38	
8.46/6.38	  1: eval2->eval2, Arg_1: 2^(max([0, 2*Arg_0])+max([0, (1+Arg_0)*max([-2, 2*Arg_0])])) {O(EXP)}
8.46/6.38	
8.46/6.38	  2: eval2->eval1, Arg_0: Arg_0 {O(n)}
8.46/6.38	
8.46/6.38	  2: eval2->eval1, Arg_1: max([1, 2^(max([0, 2*Arg_0])+max([0, (1+Arg_0)*max([-2, 2*Arg_0])]))]) {O(EXP)}
8.46/6.38	
8.46/6.38	  3: start->eval1, Arg_0: Arg_0 {O(n)}
8.46/6.38	
8.46/6.38	  3: start->eval1, Arg_1: Arg_1 {O(n)}
8.46/6.38	
8.46/6.38	
8.46/6.38	----------------------------------------
8.46/6.38	
8.46/6.38	(2)
8.46/6.38	BOUNDS(1, max(1, 3 + 2 * Arg_0) + nat(2 * Arg_0) + nat(Arg_0 * max(2 * Arg_0, -2) + max(2 * Arg_0, -2)))
8.57/6.40	EOF