Expected Cost
|
+- [f]
|    0
|
+- [Program]
|    fast :~ {1 : 10}
|    medium :~ {1 : 3}
|    slow :~ {1 : 1}
|    volMeasured :~ {1 : 0}
|    r :~ {1/2 : 0;1/2 : 1}
|    volMeasured :~ {1 : r + volMeasured}
|    0:While(volToFill ≥ volMeasured)
|        1:If(fast + volMeasured ≥ 1 + volToFill) Then
|            r :~ {1/2 : 9;1/2 : 10}
|            volMeasured :~ {1 : r + volMeasured}
|          Else
|            2:If(medium + volMeasured ≥ 1 + volToFill) Then
|                r :~ {1/3 : 2;1/3 : 3;1/3 : 4}
|                volMeasured :~ {1 : r + volMeasured}
|              Else
|                r :~ {1/3 : 0;1/3 : 1;1/3 : 2}
|                volMeasured :~ {1 : r + volMeasured}
|        r :~ {1/2 : 0;1/2 : 1}
|        volMeasured :~ {1 : r + volMeasured}
|        Tick(1)
|
+- While.step
|  |
|  +- [Problem]
|  |    0:While(volToFill ≥ volMeasured)
|  |        1:If(fast + volMeasured ≥ 1 + volToFill) Then
|  |            r :~ {1/2 : 9;1/2 : 10}
|  |            volMeasured :~ {1 : r + volMeasured}
|  |          Else
|  |            2:If(medium + volMeasured ≥ 1 + volToFill) Then
|  |                r :~ {1/3 : 2;1/3 : 3;1/3 : 4}
|  |                volMeasured :~ {1 : r + volMeasured}
|  |              Else
|  |                r :~ {1/3 : 0;1/3 : 1;1/3 : 2}
|  |                volMeasured :~ {1 : r + volMeasured}
|  |        r :~ {1/2 : 0;1/2 : 1}
|  |        volMeasured :~ {1 : r + volMeasured}
|  |        Tick(1)
|  |
|  +- [f]
|  |    0
|  |
|  +- Expected Cost Body
|  |  |
|  |  `- [1 | ⊤]
|  |
|  +- linear-template
|  |  |
|  |  `- 1 + [1 + -volMeasured + volToFill | 1 + -volMeasured + volToFill ≥ 0]
|  |
|  +- [Norms]
|  |    [[1 | ⊤],[1 + -volMeasured + volToFill | 1 + -volMeasured + volToFill ≥ 0]]
|  |
|  +- [Invariant]
|  |    volToFill ≥ volMeasured ==> [1 | ⊤] + h([1 | ⊤],ite(fast + volMeasured ≥ 1 + volToFill,1/4·[-8 + -volMeasured + volToFill | -8 + -volMeasured + volToFill ≥ 0] + 1/4·[-9 + -volMeasured + volToFill | -9 + -volMeasured + volToFill ≥ 0] + 1/4·[-9 + -volMeasured + volToFill | -9 + -volMeasured + volToFill ≥ 0] + 1/4·[-10 + -volMeasured + volToFill | -10 + -volMeasured + volToFill ≥ 0],ite(medium + volMeasured ≥ 1 + volToFill,1/6·[-1 + -volMeasured + volToFill | -1 + -volMeasured + volToFill ≥ 0] + 1/6·[-2 + -volMeasured + volToFill | -2 + -volMeasured + volToFill ≥ 0] + 1/6·[-2 + -volMeasured + volToFill | -2 + -volMeasured + volToFill ≥ 0] + 1/6·[-3 + -volMeasured + volToFill | -3 + -volMeasured + volToFill ≥ 0] + 1/6·[-3 + -volMeasured + volToFill | -3 + -volMeasured + volToFill ≥ 0] + 1/6·[-4 + -volMeasured + volToFill | -4 + -volMeasured + volToFill ≥ 0],1/6·[1 + -volMeasured + volToFill | 1 + -volMeasured + volToFill ≥ 0] + 1/6·[-volMeasured + volToFill | -volMeasured + volToFill ≥ 0] + 1/6·[-volMeasured + volToFill | -volMeasured + volToFill ≥ 0] + 1/6·[-1 + -volMeasured + volToFill | -1 + -volMeasured + volToFill ≥ 0] + 1/6·[-1 + -volMeasured + volToFill | -1 + -volMeasured + volToFill ≥ 0] + 1/6·[-2 + -volMeasured + volToFill | -2 + -volMeasured + volToFill ≥ 0]))) ≼ h([1 | ⊤],[1 + -volMeasured + volToFill | 1 + -volMeasured + volToFill ≥ 0])
|  |    volMeasured ≥ 1 + volToFill ==> 0 ≼ h([1 | ⊤],[1 + -volMeasured + volToFill | 1 + -volMeasured + volToFill ≥ 0])
|  |
|  `- 6/5·[1 + -volMeasured + volToFill | 1 + -volMeasured + volToFill ≥ 0]
|
`- 3/5·[1 + volToFill | 1 + volToFill ≥ 0] + 3/5·[volToFill | volToFill ≥ 0]