I am confused about the proof offered here: https://sicp-solutions.net/post/sicp-solution-exercise-1-13/.
The proof starts at Fib(n) = Fib(n - 1) + Fib(n - 2), which is true by definition. Then it invokes a definition we're trying to prove, namely Fib(n) = (φn - ψn)/sqrt(5), to infer the next equation. And so on until we infer that φn - ψn = φn - ψn, "[t]hus proving that Fib(n) = (φn - ψn)/sqrt(5) is true."
How does this prove that Fib(n) = (φn - ψn)/sqrt(5) is true? (If indeed the proof is valid. If it isn't valid, what would need to be changed to make it so?)
(Is part of the idea behind this presentation that since φn - ψn = φn - ψn is a tautology, we could have started from there and moved backward to some point? But what point would that be?)
This argument is incomplete. What it proves is that $G_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}$ satisfies the same recursion as the Fibonacci numbers. But to prove that it's equal to the Fibonacci numbers you also need to check that it has the same initial conditions
$$G_0 = F_0 = 0, G_1 = F_1 = 1.$$
Fortunately this is not hard.