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]