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