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