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


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WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, max(2, 4 + 2 * Arg_0) + nat(1 + Arg_1)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 455 ms]
(2) BOUNDS(1, max(2, 4 + 2 * Arg_0) + nat(1 + Arg_1))


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

The start-symbols are:[start_2]


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(1) Koat2 Proof (FINISHED)
YES( ?, 2+2*max([0, 1+Arg_0])+max([0, 1+Arg_1]) {O(n)})



Initial Complexity Problem:

  Start:  start  

  Program_Vars:  Arg_0, Arg_1  

  Temp_Vars:    

  Locations:  eval, start  

  Transitions:

  0: eval->eval

  1: eval->eval

  2: eval->eval

  3: eval->eval

  4: start->eval  



Timebounds: 

  Overall timebound: 2+2*max([0, 1+Arg_0])+max([0, 1+Arg_1]) {O(n)}

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

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

  2: eval->eval: max([0, 1+Arg_1]) {O(n)}

  3: eval->eval: 1 {O(1)}

  4: start->eval: 1 {O(1)}



Costbounds:

  Overall costbound: 2+2*max([0, 1+Arg_0])+max([0, 1+Arg_1]) {O(n)}

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

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

  2: eval->eval: max([0, 1+Arg_1]) {O(n)}

  3: eval->eval: 1 {O(1)}

  4: start->eval: 1 {O(1)}



Sizebounds:

`Lower:

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

  0: eval->eval, Arg_1: Arg_1 {O(n)}

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

  1: eval->eval, Arg_1: 1 {O(1)}

  2: eval->eval, Arg_0: min([0, Arg_0]) {O(n)}

  2: eval->eval, Arg_1: 0 {O(1)}

  3: eval->eval, Arg_0: inf {Infinity}

  3: eval->eval, Arg_1: inf {Infinity}

  4: start->eval, Arg_0: Arg_0 {O(n)}

  4: start->eval, Arg_1: Arg_1 {O(n)}

`Upper:

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

  0: eval->eval, Arg_1: Arg_1 {O(n)}

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

  1: eval->eval, Arg_1: Arg_1 {O(n)}

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

  2: eval->eval, Arg_1: Arg_1 {O(n)}

  3: eval->eval, Arg_0: -(inf) {Infinity}

  3: eval->eval, Arg_1: -(inf) {Infinity}

  4: start->eval, Arg_0: Arg_0 {O(n)}

  4: start->eval, Arg_1: Arg_1 {O(n)}


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(2)
BOUNDS(1, max(2, 4 + 2 * Arg_0) + nat(1 + Arg_1))