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