Expected Cost
|
+- [f]
|    0
|
+- [Program]
|    b :~ {1 : 0}
|    0:While(b ≥ 1)
|        Choice
|          1/2: c :~ {1 : 1}
|               b :~ {1 : 0}
|               Tick(1)
|          1/2: c :~ {1 : 0}
|               Tick(1)
|        Tick(2)
|    Tick(1)
|
+- Expected Cost
|  |
|  +- [f]
|  |    0
|  |
|  +- [Program]
|  |    0:While(b ≥ 1)
|  |        Choice
|  |          1/2: c :~ {1 : 1}
|  |               b :~ {1 : 0}
|  |               Tick(1)
|  |          1/2: c :~ {1 : 0}
|  |               Tick(1)
|  |        Tick(2)
|  |
|  +- While.step
|  |  |
|  |  +- [Problem]
|  |  |    0:While(b ≥ 1)
|  |  |        Choice
|  |  |          1/2: c :~ {1 : 1}
|  |  |               b :~ {1 : 0}
|  |  |               Tick(1)
|  |  |          1/2: c :~ {1 : 0}
|  |  |               Tick(1)
|  |  |        Tick(2)
|  |  |
|  |  +- [f]
|  |  |    0
|  |  |
|  |  +- Expected Cost Body
|  |  |  |
|  |  |  `- 3
|  |  |
|  |  +- linear-template
|  |  |  |
|  |  |  `- 1 + [b | b ≥ 0]
|  |  |
|  |  +- [Norms]
|  |  |    [[1 | ⊤],[b | b ≥ 0]]
|  |  |
|  |  +- [Invariant]
|  |  |    b ≥ 1 ==> 3 + h([1 | ⊤],1/2·[b | b ≥ 0]) ≼ h([1 | ⊤],[b | b ≥ 0])
|  |  |    1 ≥ 1 + b ==> 0 ≼ h([1 | ⊤],[b | b ≥ 0])
|  |  |
|  |  `- 6·[b | b ≥ 0]
|  |
|  `- 6·[b | b ≥ 0]
|
+- Expected Cost
|  |
|  +- [f]
|  |    0
|  |
|  +- [Program]
|  |    Tick(1)
|  |
|  `- [1 | ⊤]
|
+- Expected Cost
|  |
|  +- [f]
|  |    [1 | ⊤]
|  |
|  +- [Program]
|  |    0:While(b ≥ 1)
|  |        Choice
|  |          1/2: c :~ {1 : 1}
|  |               b :~ {1 : 0}
|  |               Tick(1)
|  |          1/2: c :~ {1 : 0}
|  |               Tick(1)
|  |        Tick(2)
|  |
|  `- [1 | ⊤]
|
`- [1 | ⊤]