/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files


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WORST_CASE(?, O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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))).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 247 ms]
(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)))


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
eval1(A, B) -> Com_1(eval2(A, 1)) :|: A >= 0
eval2(A, B) -> Com_1(eval2(A, 2 * B)) :|: A >= 0 && B >= 1 && A >= B + 1
eval2(A, B) -> Com_1(eval1(A - 1, B)) :|: A >= 0 && B >= 1 && B >= A
start(A, B) -> Com_1(eval1(A, B)) :|: TRUE

The start-symbols are:[start_2]


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(1) Koat2 Proof (FINISHED)
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)})



Initial Complexity Problem:

  Start:  start  

  Program_Vars:  Arg_0, Arg_1  

  Temp_Vars:    

  Locations:  eval1, eval2, start  

  Transitions:

  eval1(Arg_0,Arg_1) -> eval2(Arg_0,1):|:0 <= Arg_0

  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

  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

  start(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:  



Timebounds: 

  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)}

  0: eval1->eval2: max([0, 1+Arg_0]) {O(n)}

  1: eval2->eval2: max([0, 2*Arg_0])+max([0, (1+Arg_0)*max([-2, 2*Arg_0])]) {O(n^2)}

  2: eval2->eval1: max([0, 1+Arg_0]) {O(n)}

  3: start->eval1: 1 {O(1)}



Costbounds:

  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)}

  0: eval1->eval2: max([0, 1+Arg_0]) {O(n)}

  1: eval2->eval2: max([0, 2*Arg_0])+max([0, (1+Arg_0)*max([-2, 2*Arg_0])]) {O(n^2)}

  2: eval2->eval1: max([0, 1+Arg_0]) {O(n)}

  3: start->eval1: 1 {O(1)}



Sizebounds:

`Lower:

  0: eval1->eval2, Arg_0: 0 {O(1)}

  0: eval1->eval2, Arg_1: 1 {O(1)}

  1: eval2->eval2, Arg_0: 2 {O(1)}

  1: eval2->eval2, Arg_1: 2 {O(1)}

  2: eval2->eval1, Arg_0: -1 {O(1)}

  2: eval2->eval1, Arg_1: 1 {O(1)}

  3: start->eval1, Arg_0: Arg_0 {O(n)}

  3: start->eval1, Arg_1: Arg_1 {O(n)}

`Upper:

  0: eval1->eval2, Arg_0: Arg_0 {O(n)}

  0: eval1->eval2, Arg_1: 1 {O(1)}

  1: eval2->eval2, Arg_0: Arg_0 {O(n)}

  1: eval2->eval2, Arg_1: 2^(max([0, 2*Arg_0])+max([0, (1+Arg_0)*max([-2, 2*Arg_0])])) {O(EXP)}

  2: eval2->eval1, Arg_0: Arg_0 {O(n)}

  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)}

  3: start->eval1, Arg_0: Arg_0 {O(n)}

  3: start->eval1, Arg_1: Arg_1 {O(n)}


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(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)))