Expected Cost
|
+- [f]
|    0
|
+- [Program]
|    0:While(x ≥ 2)
|        Choice
|          1/3: x :~ {1 : -1 + x}
|          2/3: x :~ {1 : -2 + x}
|        Tick(1)
|
+- While.step
|  |
|  +- [Problem]
|  |    0:While(x ≥ 2)
|  |        Choice
|  |          1/3: x :~ {1 : -1 + x}
|  |          2/3: x :~ {1 : -2 + x}
|  |        Tick(1)
|  |
|  +- [f]
|  |    0
|  |
|  +- Expected Cost Body
|  |  |
|  |  `- [1 | ⊤]
|  |
|  +- linear-template
|  |  |
|  |  `- 1 + [-1 + x | -1 + x ≥ 0]
|  |
|  +- [Norms]
|  |    [[1 | ⊤],[-1 + x | -1 + x ≥ 0]]
|  |
|  +- [Invariant]
|  |    x ≥ 2 ==> [1 | ⊤] + h([1 | ⊤],1/3·[-2 + x | -2 + x ≥ 0] + 2/3·[-3 + x | -3 + x ≥ 0]) ≼ h([1 | ⊤],[-1 + x | -1 + x ≥ 0])
|  |    2 ≥ 1 + x ==> 0 ≼ h([1 | ⊤],[-1 + x | -1 + x ≥ 0])
|  |
|  `- [-1 + x | -1 + x ≥ 0]
|
`- [-1 + x | -1 + x ≥ 0]