Inference rules

Inference rules of algorithmic logic \textsc{Inference rules IR} :

\begin{trivlist} \item[$R_1$]\qquad $\dfrac{\alpha ,(\alpha \Rightarrow \beta )}{\beta }$ \item[$R_{2}$]\qquad $\dfrac{ (\alpha \Rightarrow \beta )}{(K\alpha \Rightarrow K\beta ) }$ \item[$R_{3}$]\qquad $\dfrac{\{s({\bf if}\ \gamma \ {\bf then}\ K\ {\bf fi})^{i}(\lnot \gamma \wedge \alpha )\Longrightarrow \beta \}_{i\in N}}{(s({\bf while}\ \gamma \ {\bf do}\ K\ {\bf od}\ \alpha )\Longrightarrow \beta )}$ \item[$R_{4}$]\qquad $\dfrac{\{(K^i\alpha \Longrightarrow \beta )\}_{i\in N}}{(\bigcup K\alpha \Longrightarrow \beta )}$ \item[$R_{5}$]\qquad $\dfrac{\{(\alpha \Longrightarrow K^i\beta )\}_{i\in N}}{(\alpha \Longrightarrow \bigcap K\beta )}$ \item[$R_{6}$]\qquad $\dfrac{(\alpha (x)~\Longrightarrow ~\beta )}{((\exists x)\alpha (x)~\Longrightarrow ~\beta )}$ \item[$R_{7}$]\qquad $\dfrac{(\beta ~\Longrightarrow ~\alpha (x))}{(\beta \Longrightarrow (\forall )\alpha (x))}$ \end{trivlist} In rules $R_6$ and $R_7$, it is assumed that $x$ is a variable which is not free in $ \beta $, i.e. $x \notin FV(\beta )$. The rules are known as the rule for introducing an existential quantifier into the antecedent of an implication and the rule for introducing a universal quantifier into the suc\-ces\-sor of an implication. The rules $R_4$ and $R_5$ are algorithmic counterparts of rules $R_6$ and $R_7$. They are of a different character, however, since their sets of premises are in\-finite. The rule $R_3$ for introducing a \textbf{while} into the antecedent of an implicationis of a similar nature. These three rules are called $\omega $-rules.

The rule $R_{1}$ is known as \textit{modus ponens}, or the \textit{cut} rule.

In all the above schemes of axioms and inference rules, $\alpha $, $\beta $, $\delta $ are arbi\-trary for\-mulas, $\gamma $ and $\gamma ^{\prime }$ are arbitrary open formulas, $\tau $ is an arbitrary term, $s$ is a finite se\-quence of assignment instructions, and $K$ and $M$ are arbitrary programs.