Expected Cost
|
+- [f]
|    0
|
+- [Program]
|    x1 :~ {1 : n}
|    0:While(x1 ≥ 1)
|        r :~ {1/4 : -2;1/4 : -1;1/4 : 0;1/4 : 1}
|        x1 :~ {1 : r + x1}
|        Tick(1)
|
+- While.step
|  |
|  +- [Problem]
|  |    0:While(x1 ≥ 1)
|  |        r :~ {1/4 : -2;1/4 : -1;1/4 : 0;1/4 : 1}
|  |        x1 :~ {1 : r + x1}
|  |        Tick(1)
|  |
|  +- [f]
|  |    0
|  |
|  +- Expected Cost Body
|  |  |
|  |  `- [1 | ⊤]
|  |
|  +- linear-template
|  |  |
|  |  `- 1 + [x1 | x1 ≥ 0]
|  |
|  +- [Norms]
|  |    [[1 | ⊤],[x1 | x1 ≥ 0]]
|  |
|  +- [Invariant]
|  |    x1 ≥ 1 ==> [1 | ⊤] + h([1 | ⊤],1/4·[-2 + x1 | -2 + x1 ≥ 0] + 1/4·[-1 + x1 | -1 + x1 ≥ 0] + 1/4·[x1 | x1 ≥ 0] + 1/4·[1 + x1 | 1 + x1 ≥ 0]) ≼ h([1 | ⊤],[x1 | x1 ≥ 0])
|  |    1 ≥ 1 + x1 ==> 0 ≼ h([1 | ⊤],[x1 | x1 ≥ 0])
|  |
|  `- 4·[x1 | x1 ≥ 0]
|
`- 4·[n | n ≥ 0]