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