Dynamic programming: minimum coins

Unlimited positive sizes; return fewest coins totaling amount or -1 if impossible. amount 0 returns 0 even with no sizes. Negative amount or any size<=0 throws invalid_argument. Prototype uncached recursion; final O(amount*size_count) DP/O(amount) memory. Preserve sizes.

Public signatures:

Int prototype(const Vec<Int>& sizes,const Int& amount);
Int solve(const Vec<Int>& sizes,const Int& amount);

Compile and run from this directory:

c++ -std=c++20 -O2 -Wall -Wextra -pedantic -pthread main.cpp -o practice
./practice --stage prototype
./practice --stage final

Edit exercise.hpp. Both stages start unfinished and must report failures until implemented. Keep main.cpp as the executable contract. The runner returns nonzero for a failed check, an unexpected exception, an invalid stage, or an empty suite. C++20 and only the standard library are required.