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