proof-of-correctness

I'm trying to get a fuller picture of the use of the optimal substructure property in dynamic programming, yet I've gone blind on why we have to prove that any optimal solution to the problem contains within it optimal solutions to the sub-problems. Wouldn't it be enough to show that some optimal solution to the problem has this property, and then use this to argue that since the solution built by our recursive algorithm is at least as good as an optimal solution, it will itself be optimal? In other words, I fail to spot where in the correctness argument for our algorithm we need that all

I am new to lambda calculus and struggling to prove the following. SKK and II are beta equivalent. where S = lambda xyz.xz(yz) K = lambda xy.x I = lambda x.x I tried to beta reduce SKK by opening it up, but got nowhere, it becomes to messy. Dont think SKK can be reduced further without expanding S, K. SKK = (λxyz.xz(yz))KK → λz.Kz(Kz) (in two steps actually, for the two parameters) Kz = (λxy.x)z → λy.z λz.Kz(Kz) → λz.(λy.z)(λy.z) (again, several steps) → λz.z = I (You should be able to prove that II → I ) ;another approach with fewer steps, first reduce SK to λyz.z; SKK = (λxyz.xz(yz))KK → λyz

I was thinking about the fact that we can prove a program has bugs. We can test it to assess that it is more or less bug resistant. But is there a way (even theoretically) to prove that a program has no bug ? For simple programs, such as a "Hello World", I guess we should be able to do it. But what about larger programs ? There are nowadays many different formalisms that can be used to prove programs correct (e.g., formalizations in proof assistants, dependently typed programming languages, separation logic, ...). As noted by others, there is no automatic way to prove any given program correct

Most proof assistants are functional programming languages with dependent types. They can proof programs/algorithms. I'm interested, instead, in proof assistant suitable best for mathematics and only (calculus for instance). Can you recommend one? I heard about Mizar but I don’t like that the source code is closed, but if it is best for math I will use it. How well the new languages such as Agda and Idris are suited for mathematical proofs? Coq has extensive libraries covering real analysis. Various developments come to mind: the standard library and projects building on it such as the now