User:Physis/Gödel-Herbrand-Kleene equational calculus

Definition
Defined in the follwing way.

The calculus consists of a set (or system) of equations. An equation is of form =, where the left-hand side term may be required to contain a principial letter. Let the set of nonlogical symbols contain the unary f (besides the arithmetical ones 0 and s). This f is just an example of a particular signature, in generally, an arbitrary signature must be considered (which contains the signature of natural numbers at least as an 0 and s).

Syntax



 * ::= 0
 * ::= s
 * ::= f
 * ::= g



Rules

 * $$\frac{\gamma \in \Gamma}{\Gamma \vdash \gamma}$$


 * $$\frac{\Gamma \vdash \phi}{\Gamma \vdash \phi[x := \mathbf m]}$$


 * $$\frac{\Gamma \vdash \sigma = \tau, m \in \mathbb N, n \in \mathbb N, \Gamma \vdash f \mathbf m = \mathbf n}{\Gamma \vdash \sigma = \tau[f \mathbf m \to_1 \mathbf n]}$$

Expressing power
Equivalent power with the theory of partial recursive functions.