ExecutionLog | `- Expected Cost | +- [f] | 0 | +- [Program] | i :~ {1 : 1} | 0:While(n 1 + i) | j :~ {1 : -1 + i} | 1:While(j 1) | Choice | 1/3: j :~ {1 : -1 + j} | 2/3: j :~ {1 : j} | Tick(1) | i :~ {1 : 1 + i} | +- While.step | | | +- [Problem] | | 0:While(n 1 + i) | | j :~ {1 : -1 + i} | | 1:While(j 1) | | Choice | | 1/3: j :~ {1 : -1 + j} | | 2/3: j :~ {1 : j} | | Tick(1) | | i :~ {1 : 1 + i} | | | +- [f] | | 0 | | | +- Expected Cost Body | | | | | +- Expected Cost | | | | | | | +- [f] | | | | 0 | | | | | | | +- [Program] | | | | 1:While(j 1) | | | | Choice | | | | 1/3: j :~ {1 : -1 + j} | | | | 2/3: j :~ {1 : j} | | | | Tick(1) | | | | | | | +- While.step | | | | | | | | | +- [Problem] | | | | | 1:While(j 1) | | | | | Choice | | | | | 1/3: j :~ {1 : -1 + j} | | | | | 2/3: j :~ {1 : j} | | | | | Tick(1) | | | | | | | | | +- [f] | | | | | 0 | | | | | | | | | +- Expected Cost Body | | | | | | | | | | | `- [1 | ] | | | | | | | | | +- linear-template | | | | | | | | | | | `- 1 + [j | j 0] | | | | | | | | | +- [Norms] | | | | | [[1 | ],[j | j 0]] | | | | | | | | | +- [Invariant] | | | | | j 1 ==> [1 | ] + h([1 | ],1/3[-1 + j | -1 + j 0] + 2/3[j | j 0]) h([1 | ],[j | j 0]) | | | | | 1 1 + j ==> 0 h([1 | ],[j | j 0]) | | | | | | | | | `- 3[j | j 0] | | | | | | | `- 3[j | j 0] | | | | | +- Expected Cost | | | | | | | +- [f] | | | | 0 | | | | | | | +- [Program] | | | | i :~ {1 : 1 + i} | | | | | | | `- 0 | | | | | +- Expected Cost | | | | | | | +- [f] | | | | 0 | | | | | | | +- [Program] | | | | 1:While(j 1) | | | | Choice | | | | 1/3: j :~ {1 : -1 + j} | | | | 2/3: j :~ {1 : j} | | | | Tick(1) | | | | | | | `- 0 | | | | | `- 3[-1 + i | -1 + i 0] | | | +- mixed-lin-template | | | | | `- 1 + [-1 + i | -1 + i 0] + [-1 + i | -1 + i 0][-i + n | -i + n 0] + 2([-i + n | -i + n 0]) + [-i + n | -i + n 0]^2 | | | +- [Norms] | | [[1 | ],[-1 + i | -1 + i 0],[i + in + -i^2 + -n | -1 + i 0 -i + n 0],[-i + n | -i + n 0],[-2(in) + i^2 + n^2 | -i + n 0]] | | | +- [Invariant] | | n 1 + i ==> 3[-1 + i | -1 + i 0] + h([1 | ],[i | i 0],[-i + in + -i^2 | -1 + -i + n 0 i 0],[-1 + -i + n | -1 + -i + n 0],[1 + 2(i) + -2(in) + i^2 + -2(n) + n^2 | -1 + -i + n 0]) h([1 | ],[-1 + i | -1 + i 0],[i + in + -i^2 + -n | -1 + i 0 -i + n 0],[-i + n | -i + n 0],[-2(in) + i^2 + n^2 | -i + n 0]) | | 1 + i 1 + n ==> 0 h([1 | ],[-1 + i | -1 + i 0],[i + in + -i^2 + -n | -1 + i 0 -i + n 0],[-i + n | -i + n 0],[-2(in) + i^2 + n^2 | -i + n 0]) | | | `- 3[i + in + -i^2 + -n | -1 + i 0 -i + n 0] + 3/2[-2(in) + i^2 + n^2 | -i + n 0] | `- 3/2[1 + -2(n) + n^2 | -1 + n 0] [Success] 3/2[1 + -2(n) + n^2 | -1 + n 0] Degree: 2