Let me explain what is going on here by way of an example:
\psi(p_0,p_1,p) = p \leftrightarrow (p_0 \land p_1).
That is, $\psi(p_0,p_1,p)$ is true iff $p = p_0 \land p_1$.
If you define
\phi(p_0,p_1) = p_0 \land p_1,
then we indeed have
\psi(p_0,p_1,p) \to (\phi(p_0,p_1) \leftrightarrow p),
which means that if $\psi(p_0,p_1,p)$ holds then $...
This problem sounds quite abstract and I'll sketch a proof below. To prove the final material conditional is a tautology, we only need to prove $\psi(p_0, \ldots p_n, p) \wedge \psi(p_0, \ldots p_n, q) \land \lnot (p \leftrightarrow q)$ is unsatisfiable, which means it can never be the case that $p,q$ can have opposite truth values while both $\psi(p_0, \...
I'll add some steps to make it very clear. You want to prove q.
You can take any statement s, then say "either s or not s is true", and then prove "if s is true then q is true" and "if not s is true then q is true". That happens actually quite often; you might have the statement "x ≤ 1" and you can prove that if x ≤ 1 ...
There is no problem with assuming $ \neg q$ towards a contradiction and showing that this implies $q$.
Here is an example, adapted from Euclid's proof that there are infinitely many primes.
Let $p$ be a prime number and suppose we want to prove "there exists a prime number larger than $p$".
We proceed by contradiction, i.e., we assume that "...
We like conjunctive normal form because all circuits can be transformed into conjunctive normal from in linear time via the Tseitin Transformation. It's a clean, normalized data structure for solver implementation. CNF conversion without Tseitin variables may lead to exponential expansion.
That's the standard answer, but, as you see, there are nuances. ...
Conjunctive normal form first appears, in this context, in Davis and Putnam's A computing procedure for quantification theory, in which they describe a primitive form of the DPLL algorithm (which appears in a follow-up work of Davis, Logemann, and Loveland, and is the basis of all modern SAT solvers). They explain that one key property of CNF is that any ...
It's because this is close to how real-world problems tend to be expressed.
SAT has a lot of practical applications, from controlling power grids to integrated circuit layout. In pretty much every case, the problem is that you have a bunch of complex constraints that all need to be satisfied.
However you express each constraint, they ultimately get joined ...
First of all in your above 3 "recursive definitions" linking $\Sigma$ with $\Sigma_2$, clearly the function $g$ in $\Sigma_2$ doesn't have any role in defining the values of $f(t)$ in $\Sigma$, the values of $f(t)$ should be already defined in language $\Sigma$.
To check your 3 "recursive definitions" more deeply, it seems here the map ...