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