Commit b62dbf41 by Ulrich Rabenstein

### added some definitions / theorems

parent f60b52c3
 ... ... @@ -12,6 +12,14 @@ \usepackage{wrapfig} \usepackage[utf8]{inputenc} \renewcommand{\baselinestretch}{.99} \newcommand{\KIbox}{\text{K}_\Box^{\text{I}}} \newcommand{\KClbox}{\text{K}_\Box^{\text{Cl}}} \newcommand{\adequate}[2]{ \forall A, \KClbox \oplus #1 \vdash A \Leftrightarrow \KIbox \oplus #1 \vdash A^{#2}} \newcommand{\kuroda}[1]{\Box \dneg #1 \rightarrow \dneg \Box #1} \newcommand{\dneg}{{\neg\neg}} \title[DNegMod]{Double-Negation Translation of Intuitionistic Modal Logics in Coq} \author{Miriam Polzer \& Ulrich Rabenstein} ... ... @@ -48,17 +56,48 @@ \begin{frame} Ulrich: \\ adequate eq ggr \begin{theorem} For $t$ in the triangle: \\ $(\adequate Z t) \Leftrightarrow \KIbox \oplus Z \vdash Z^t$ \\ \end{theorem} \begin{proof} $\rightarrow$ easy \\ $\leftarrow$ complicated syntactic proof... \end{proof} \end{frame} \begin{frame} envelopes \\ evtl. beispiele \\ theorem 5 \begin{definition} A is a pre-envelope iff $\forall Z, \KIbox \oplus Z \vdash \dneg sub_\dneg(A) \rightarrow A^{kol}$ \\ A is a post-envelope iff $\forall Z, \KIbox \oplus Z \vdash A^{kol} \rightarrow \dneg sub_\dneg(A)$ \\ A is a $\dneg$-envelope iff $\forall Z, \KIbox \oplus Z \vdash A^{kol} \leftrightarrow \dneg sub_\dneg(A)$ \\ \end{definition} %evtl. beispiele von envelopes \begin{theorem} Let B be a post-envelope and C be a pre-envelope then $\adequate {(B \rightarrow C)} t$. \end{theorem} \begin{Proof} follows directly from the definition of envelopes \end{Proof} \end{frame} \begin{frame} theorem 6 \title{Glivenko-Translation} \begin{definition} $A^{glv} = \dneg A$ \\ Kuroda-axiom $\; \kuroda A$ \end{definition} Assuming Kuroda-axiom, $glv$ becomes equivalent to the ones in the triangle. Therefore: \begin{theorem} $\KIbox \oplus Z \vdash \kuroda A \; \Rightarrow$ \\ $\adequate Z {glv}$ \end{theorem} \end{frame} \begin{frame} ... ...
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