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