Expected Cost | +- [f] | 0 | +- [Program] | 0:While(l ≥ 8) | l :~ {1 : -8 + l} | Tick(n) | 1:While(l ≥ 1) | l :~ {1 : -1 + l} | Tick(1) | +- Expected Cost | | | +- [f] | | 0 | | | +- [Program] | | 0:While(l ≥ 8) | | l :~ {1 : -8 + l} | | Tick(n) | | | +- While.step | | | | | +- [Problem] | | | 0:While(l ≥ 8) | | | l :~ {1 : -8 + l} | | | Tick(n) | | | | | +- [f] | | | 0 | | | | | +- Expected Cost Body | | | | | | | `- [n | n ≥ 0] | | | | | +- mixed-lin-template | | | | | | | `- 1 + 2·([-7 + l | -7 + l ≥ 0]) + [-7 + l | -7 + l ≥ 0]·[n | n ≥ 0] + [-7 + l | -7 + l ≥ 0]^2 + [n | n ≥ 0] | | | | | +- [Norms] | | | [[1 | ⊤],[-7 + l | -7 + l ≥ 0],[l·n + -7·(n) | -7 + l ≥ 0 ∧ n ≥ 0],[49 + -14·(l) + l^2 | -7 + l ≥ 0],[n | n ≥ 0]] | | | | | +- [Invariant] | | | l ≥ 8 ==> [n | n ≥ 0] + h([1 | ⊤],[-15 + l | -15 + l ≥ 0],[l·n + -15·(n) | -15 + l ≥ 0 ∧ n ≥ 0],[225 + -30·(l) + l^2 | -15 + l ≥ 0],[n | n ≥ 0]) ≼ h([1 | ⊤],[-7 + l | -7 + l ≥ 0],[l·n + -7·(n) | -7 + l ≥ 0 ∧ n ≥ 0],[49 + -14·(l) + l^2 | -7 + l ≥ 0],[n | n ≥ 0]) | | | 8 ≥ 1 + l ==> 0 ≼ h([1 | ⊤],[-7 + l | -7 + l ≥ 0],[l·n + -7·(n) | -7 + l ≥ 0 ∧ n ≥ 0],[49 + -14·(l) + l^2 | -7 + l ≥ 0],[n | n ≥ 0]) | | | | | `- [l·n + -7·(n) | -7 + l ≥ 0 ∧ n ≥ 0] | | | `- [l·n + -7·(n) | -7 + l ≥ 0 ∧ n ≥ 0] | +- Expected Cost | | | +- [f] | | 0 | | | +- [Program] | | 1:While(l ≥ 1) | | l :~ {1 : -1 + l} | | Tick(1) | | | +- While.step | | | | | +- [Problem] | | | 1:While(l ≥ 1) | | | l :~ {1 : -1 + l} | | | Tick(1) | | | | | +- [f] | | | 0 | | | | | +- Expected Cost Body | | | | | | | `- [1 | ⊤] | | | | | +- linear-template | | | | | | | `- 1 + [l | l ≥ 0] | | | | | +- [Norms] | | | [[1 | ⊤],[l | l ≥ 0]] | | | | | +- [Invariant] | | | l ≥ 1 ==> [1 | ⊤] + h([1 | ⊤],[-1 + l | -1 + l ≥ 0]) ≼ h([1 | ⊤],[l | l ≥ 0]) | | | 1 ≥ 1 + l ==> 0 ≼ h([1 | ⊤],[l | l ≥ 0]) | | | | | `- [l | l ≥ 0] | | | `- [l | l ≥ 0] | +- Expected Cost | | | +- [f] | | [l | l ≥ 0] | | | +- [Program] | | 0:While(l ≥ 8) | | l :~ {1 : -8 + l} | | Tick(n) | | | +- While.step | | | | | +- [Problem] | | | 0:While(l ≥ 8) | | | l :~ {1 : -8 + l} | | | Tick(n) | | | | | +- [f] | | | [l | l ≥ 0] | | | | | +- linear-template | | | | | | | `- 1 + 2·([-7 + l | -7 + l ≥ 0]) + [l | l ≥ 0] | | | | | +- [Norms] | | | [[1 | ⊤],[-7 + l | -7 + l ≥ 0],[l | l ≥ 0]] | | | | | +- [Invariant] | | | l ≥ 8 ==> 0 + h([1 | ⊤],[-15 + l | -15 + l ≥ 0],[-8 + l | -8 + l ≥ 0]) ≼ h([1 | ⊤],[-7 + l | -7 + l ≥ 0],[l | l ≥ 0]) | | | 8 ≥ 1 + l ==> [l | l ≥ 0] ≼ h([1 | ⊤],[-7 + l | -7 + l ≥ 0],[l | l ≥ 0]) | | | | | `- 7 | | | `- 7 | `- [l·n + -7·(n) | -7 + l ≥ 0 ∧ n ≥ 0] + 7