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