Expected Cost
|
+- [f]
|    0
|
+- [Program]
|    x :~ {1 : 0}
|    0:While(n ≥ 1 + x)
|        Tick(1)
|        x :~ {1 : 1 + x}
|        break :~ {1 : 0}
|        1:While(0 ≥ break ∧  break ≥ 0 ∧  n ≥ 1 + x)
|            NonDet
|              {break :~ {1 : 1}}
|              {x :~ {1 : 1 + x}
|              Tick(1)}
|
+- While.step
|  |
|  +- [Problem]
|  |    0:While(n ≥ 1 + x)
|  |        Tick(1)
|  |        x :~ {1 : 1 + x}
|  |        break :~ {1 : 0}
|  |        1:While(0 ≥ break ∧  break ≥ 0 ∧  n ≥ 1 + x)
|  |            NonDet
|  |              {break :~ {1 : 1}}
|  |              {x :~ {1 : 1 + x}
|  |              Tick(1)}
|  |
|  +- [f]
|  |    0
|  |
|  +- Expected Cost Body
|  |  |
|  |  +- While.step
|  |  |  |
|  |  |  +- [Problem]
|  |  |  |    1:While(0 ≥ break ∧  break ≥ 0 ∧  n ≥ 1 + x)
|  |  |  |        NonDet
|  |  |  |          {break :~ {1 : 1}}
|  |  |  |          {x :~ {1 : 1 + x}
|  |  |  |          Tick(1)}
|  |  |  |
|  |  |  +- [f]
|  |  |  |    0
|  |  |  |
|  |  |  +- Expected Cost Body
|  |  |  |  |
|  |  |  |  `- [1 | ⊤]
|  |  |  |
|  |  |  +- mixed-iteration-template
|  |  |  |  |
|  |  |  |  `- [1 + -break | 1 + -break ≥ 0] + 2·([1 + -break | 1 + -break ≥ 0]·[1 + break | 1 + break ≥ 0]) + 2·([1 + -break | 1 + -break ≥ 0]·[n + -x | n + -x ≥ 0]) + [1 + -break | 1 + -break ≥ 0]^2 + [1 + break | 1 + break ≥ 0] + 2·([1 + break | 1 + break ≥ 0]·[n + -x | n + -x ≥ 0]) + [1 + break | 1 + break ≥ 0]^2 + [n + -x | n + -x ≥ 0] + [n + -x | n + -x ≥ 0]^2
|  |  |  |
|  |  |  +- [Norms]
|  |  |  |    [[1 + -break | 1 + -break ≥ 0],[1 + -break^2 | 1 + -break ≥ 0 ∧  1 + break ≥ 0],[-break·n + break·x + n + -x | 1 + -break ≥ 0 ∧  n + -x ≥ 0],[1 + -2·(break) + break^2 | 1 + -break ≥ 0],[1 + break | 1 + break ≥ 0],[break·n + -break·x + n + -x | 1 + break ≥ 0 ∧  n + -x ≥ 0],[1 + 2·(break) + break^2 | 1 + break ≥ 0],[n + -x | n + -x ≥ 0],[-2·(n·x) + n^2 + x^2 | n + -x ≥ 0]]
|  |  |  |
|  |  |  +- [Invariant]
|  |  |  |    0 ≥ break ∧  break ≥ 0 ∧  n ≥ 1 + x ==> [1 | ⊤] + h([1 + -break | 1 + -break ≥ 0],[1 + -break^2 | 1 + -break ≥ 0 ∧  1 + break ≥ 0],[-1 + break + -break·n + break·x + n + -x | -1 + n + -x ≥ 0 ∧  1 + -break ≥ 0],[1 + -2·(break) + break^2 | 1 + -break ≥ 0],sup(2,[1 + break | 1 + break ≥ 0]),sup(2·[n + -x | n + -x ≥ 0],[-1 + -break + break·n + -break·x + n + -x | -1 + n + -x ≥ 0 ∧  1 + break ≥ 0]),sup(4,[1 + 2·(break) + break^2 | 1 + break ≥ 0]),sup([n + -x | n + -x ≥ 0],[-1 + n + -x | -1 + n + -x ≥ 0]),sup([-2·(n·x) + n^2 + x^2 | n + -x ≥ 0],[1 + -2·(n) + -2·(n·x) + n^2 + 2·(x) + x^2 | -1 + n + -x ≥ 0])) ≼ h([1 + -break | 1 + -break ≥ 0],[1 + -break^2 | 1 + -break ≥ 0 ∧  1 + break ≥ 0],[-break·n + break·x + n + -x | 1 + -break ≥ 0 ∧  n + -x ≥ 0],[1 + -2·(break) + break^2 | 1 + -break ≥ 0],[1 + break | 1 + break ≥ 0],[break·n + -break·x + n + -x | 1 + break ≥ 0 ∧  n + -x ≥ 0],[1 + 2·(break) + break^2 | 1 + break ≥ 0],[n + -x | n + -x ≥ 0],[-2·(n·x) + n^2 + x^2 | n + -x ≥ 0])
|  |  |  |    0 ≥ 1 + break ∨  1 + x ≥ 1 + n ∨  break ≥ 1 ==> 0 ≼ h([1 + -break | 1 + -break ≥ 0],[1 + -break^2 | 1 + -break ≥ 0 ∧  1 + break ≥ 0],[-break·n + break·x + n + -x | 1 + -break ≥ 0 ∧  n + -x ≥ 0],[1 + -2·(break) + break^2 | 1 + -break ≥ 0],[1 + break | 1 + break ≥ 0],[break·n + -break·x + n + -x | 1 + break ≥ 0 ∧  n + -x ≥ 0],[1 + 2·(break) + break^2 | 1 + break ≥ 0],[n + -x | n + -x ≥ 0],[-2·(n·x) + n^2 + x^2 | n + -x ≥ 0])
|  |  |  |
|  |  |  `- [-break·n + break·x + n + -x | 1 + -break ≥ 0 ∧  n + -x ≥ 0]
|  |  |
|  |  `- [1 | ⊤] + [-1 + n + -x | -1 + n + -x ≥ 0]
|  |
|  +- mixed-iteration-template
|  |  |
|  |  `- [n + -x | n + -x ≥ 0] + [n + -x | n + -x ≥ 0]^2
|  |
|  +- [Norms]
|  |    [[n + -x | n + -x ≥ 0],[-2·(n·x) + n^2 + x^2 | n + -x ≥ 0]]
|  |
|  +- While.step
|  |  |
|  |  +- [Problem]
|  |  |    1:While(0 ≥ break ∧  break ≥ 0 ∧  n ≥ 1 + x)
|  |  |        NonDet
|  |  |          {break :~ {1 : 1}}
|  |  |          {x :~ {1 : 1 + x}
|  |  |          Tick(1)}
|  |  |
|  |  +- [f]
|  |  |    [-2·(n·x) + n^2 + x^2 | n + -x ≥ 0]
|  |  |
|  |  +- linear-template
|  |  |  |
|  |  |  `- 1 + [1 + -break | 1 + -break ≥ 0] + [1 + -break | 1 + -break ≥ 0]·[1 + 2·(x) | 1 + 2·(x) ≥ 0] + [1 + break | 1 + break ≥ 0] + [1 + break | 1 + break ≥ 0]·[1 + 2·(x) | 1 + 2·(x) ≥ 0] + [1 + 2·(x) | 1 + 2·(x) ≥ 0]·[n + -x | n + -x ≥ 0] + [n + -x | n + -x ≥ 0] + [-2·(n·x) + n^2 + x^2 | n + -x ≥ 0]
|  |  |
|  |  +- [Norms]
|  |  |    [[1 | ⊤],[1 + -break | 1 + -break ≥ 0],[1 + -break + -2·(break·x) + 2·(x) | 1 + -break ≥ 0 ∧  1 + 2·(x) ≥ 0],[1 + break | 1 + break ≥ 0],[1 + break + 2·(break·x) + 2·(x) | 1 + break ≥ 0 ∧  1 + 2·(x) ≥ 0],[n + 2·(n·x) + -x + -2·(x^2) | 1 + 2·(x) ≥ 0 ∧  n + -x ≥ 0],[n + -x | n + -x ≥ 0],[-2·(n·x) + n^2 + x^2 | n + -x ≥ 0]]
|  |  |
|  |  +- [Invariant]
|  |  |    0 ≥ break ∧  break ≥ 0 ∧  n ≥ 1 + x ==> 0 + h([1 | ⊤],[1 + -break | 1 + -break ≥ 0],[3 + -3·(break) + -2·(break·x) + 2·(x) | 1 + -break ≥ 0 ∧  3 + 2·(x) ≥ 0],sup(2,[1 + break | 1 + break ≥ 0]),sup(2·[1 + 2·(x) | 1 + 2·(x) ≥ 0],[3 + 3·(break) + 2·(break·x) + 2·(x) | 1 + break ≥ 0 ∧  3 + 2·(x) ≥ 0]),sup([n + 2·(n·x) + -x + -2·(x^2) | 1 + 2·(x) ≥ 0 ∧  n + -x ≥ 0],[-3 + 3·(n) + 2·(n·x) + -5·(x) + -2·(x^2) | -1 + n + -x ≥ 0 ∧  3 + 2·(x) ≥ 0]),sup([n + -x | n + -x ≥ 0],[-1 + n + -x | -1 + n + -x ≥ 0]),sup([-2·(n·x) + n^2 + x^2 | n + -x ≥ 0],[1 + -2·(n) + -2·(n·x) + n^2 + 2·(x) + x^2 | -1 + n + -x ≥ 0])) ≼ h([1 | ⊤],[1 + -break | 1 + -break ≥ 0],[1 + -break + -2·(break·x) + 2·(x) | 1 + -break ≥ 0 ∧  1 + 2·(x) ≥ 0],[1 + break | 1 + break ≥ 0],[1 + break + 2·(break·x) + 2·(x) | 1 + break ≥ 0 ∧  1 + 2·(x) ≥ 0],[n + 2·(n·x) + -x + -2·(x^2) | 1 + 2·(x) ≥ 0 ∧  n + -x ≥ 0],[n + -x | n + -x ≥ 0],[-2·(n·x) + n^2 + x^2 | n + -x ≥ 0])
|  |  |    0 ≥ 1 + break ∨  1 + x ≥ 1 + n ∨  break ≥ 1 ==> [-2·(n·x) + n^2 + x^2 | n + -x ≥ 0] ≼ h([1 | ⊤],[1 + -break | 1 + -break ≥ 0],[1 + -break + -2·(break·x) + 2·(x) | 1 + -break ≥ 0 ∧  1 + 2·(x) ≥ 0],[1 + break | 1 + break ≥ 0],[1 + break + 2·(break·x) + 2·(x) | 1 + break ≥ 0 ∧  1 + 2·(x) ≥ 0],[n + 2·(n·x) + -x + -2·(x^2) | 1 + 2·(x) ≥ 0 ∧  n + -x ≥ 0],[n + -x | n + -x ≥ 0],[-2·(n·x) + n^2 + x^2 | n + -x ≥ 0])
|  |  |
|  |  `- [-2·(n·x) + n^2 + x^2 | n + -x ≥ 0]
|  |
|  +- [Invariant]
|  |    n ≥ 1 + x ==> [1 | ⊤] + [-1 + n + -x | -1 + n + -x ≥ 0] + h([-1 + n + -x | -1 + n + -x ≥ 0],[1 + -2·(n) + -2·(n·x) + n^2 + 2·(x) + x^2 | -1 + n + -x ≥ 0]) ≼ h([n + -x | n + -x ≥ 0],[-2·(n·x) + n^2 + x^2 | n + -x ≥ 0])
|  |    1 + x ≥ 1 + n ==> 0 ≼ h([n + -x | n + -x ≥ 0],[-2·(n·x) + n^2 + x^2 | n + -x ≥ 0])
|  |
|  `- 1/2·[n + -x | n + -x ≥ 0] + 1/2·[-2·(n·x) + n^2 + x^2 | n + -x ≥ 0]
|
`- 1/2·[n | n ≥ 0] + 1/2·[n^2 | n ≥ 0]