Expected Cost
|
+- [f]
|    0
|
+- [Program]
|    0:While(x ≥ 1 + y)
|        Tick(1)
|        x :~ {1 : -1 + x + -y}
|        z :~ {1 : 100 + 2·(y)}
|        1:While(z ≥ 1)
|            Tick(1)
|            z :~ {1 : -1 + z}
|
+- While.step
|  |
|  +- [Problem]
|  |    0:While(x ≥ 1 + y)
|  |        Tick(1)
|  |        x :~ {1 : -1 + x + -y}
|  |        z :~ {1 : 100 + 2·(y)}
|  |        1:While(z ≥ 1)
|  |            Tick(1)
|  |            z :~ {1 : -1 + z}
|  |
|  +- [f]
|  |    0
|  |
|  +- Expected Cost Body
|  |  |
|  |  +- While.step
|  |  |  |
|  |  |  +- [Problem]
|  |  |  |    1:While(z ≥ 1)
|  |  |  |        Tick(1)
|  |  |  |        z :~ {1 : -1 + z}
|  |  |  |
|  |  |  +- [f]
|  |  |  |    0
|  |  |  |
|  |  |  +- Expected Cost Body
|  |  |  |  |
|  |  |  |  `- [1 | ⊤]
|  |  |  |
|  |  |  +- linear-template
|  |  |  |  |
|  |  |  |  `- 1 + [z | z ≥ 0]
|  |  |  |
|  |  |  +- [Norms]
|  |  |  |    [[1 | ⊤],[z | z ≥ 0]]
|  |  |  |
|  |  |  +- [Invariant]
|  |  |  |    z ≥ 1 ==> [1 | ⊤] + h([1 | ⊤],[-1 + z | -1 + z ≥ 0]) ≼ h([1 | ⊤],[z | z ≥ 0])
|  |  |  |    1 ≥ 1 + z ==> 0 ≼ h([1 | ⊤],[z | z ≥ 0])
|  |  |  |
|  |  |  `- [z | z ≥ 0]
|  |  |
|  |  `- [1 | ⊤] + 2·[50 + y | 100 + 2·(y) ≥ 0]
|  |
|  +- shift-avg-template
|  |  |
|  |  `- 1 + [1 + x | 1 + x ≥ 0] + [1 + y | 1 + y ≥ 0] + [50 + y | 50 + y ≥ 0]
|  |
|  +- [Norms]
|  |    [[1 | ⊤],[1 + x | 1 + x ≥ 0],[1 + y | 1 + y ≥ 0],[50 + y | 50 + y ≥ 0]]
|  |
|  +- [Invariant]
|  |    x ≥ 1 + y ∧  y ≥ 0 ==> [1 | ⊤] + 2·[50 + y | 100 + 2·(y) ≥ 0] + h([1 | ⊤],[x + -y | x + -y ≥ 0],[1 + y | 1 + y ≥ 0],[50 + y | 50 + y ≥ 0]) ≼ h([1 | ⊤],[1 + x | 1 + x ≥ 0],[1 + y | 1 + y ≥ 0],[50 + y | 50 + y ≥ 0])
|  |    1 + y ≥ 1 + x ∧  y ≥ 0 ==> 0 ≼ h([1 | ⊤],[1 + x | 1 + x ≥ 0],[1 + y | 1 + y ≥ 0],[50 + y | 50 + y ≥ 0])
|  |
|  `- [y ≥ 0] · 101·[1 + x | 1 + x ≥ 0]
|
`- [y ≥ 0] · 101·[1 + x | 1 + x ≥ 0]