Commit 77cbb5d4 by Ulrich Rabenstein

### minor

parent 5b9180c4
 ... ... @@ -52,19 +52,37 @@ äquivalenz von adeq. (mit Bild?) \end{frame} \begin{frame} Ulrich: \\ \begin{theorem} For $t$ in the triangle: \\ \begin{frame}{Reductiontheorem} \begin{theorem}[Reductiontheorem] 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... \begin{itemize} \item [$\Rightarrow$] Obviously $\KClbox \oplus Z \vdash Z$, by the premise $\KIbox \oplus Z \vdash Z^t$. \item [$\Leftarrow$] Let $\KIbox \oplus Z \vdash Z^t$. \begin{itemize} \item [$\leftarrow$] Let $\KIbox \oplus Z \vdash A^t$, then $\KClbox \oplus Z \vdash A^t$ and thus $\KClbox \oplus Z \vdash A$. \item [$\rightarrow$] Induction on $\KClbox \oplus Z \vdash A$. \\ On the blackboard... %Box-case: Premises: \\ % $\forall A \in F, \KIbox \oplus Z \vdash G^t \Rightarrow \dneg \Box A^t$ and \\ % $\KIbox \oplus Z \vdash F^t \Rightarrow h^t$. \\ %From the first one: $\forall A \in F^t, \KIbox \oplus Z \vdash G^t \Rightarrow \dneg \Box A$. \\ %By the premises and a modified box-rule, we get $\KIbox \oplus Z \vdash G^t \Rightarrow \dneg \Box B^t$. \end{itemize} \end{itemize} \end{proof} \end{frame} \begin{frame} \begin{frame}{Envelopes} \begin{definition} \vspace{.3cm} {\centering ... ... @@ -89,12 +107,12 @@ \end{frame} \begin{frame} \begin{frame}{Adequatness conditions} \begin{theorem} \begin{itemize} \item Let B be a post-envelope and C be a pre-envelope then $\adequate {(B \rightarrow C)} t$. \item Let A be a $\dneg$-envelope or Z be a Kuroda-envelope, then $\adequate Z t$. \item Let Z be a $\dneg$-envelope or a Kuroda-envelope, then $\adequate Z t$. \end{itemize} ... ... @@ -104,22 +122,26 @@ \end{Proof} \end{frame} \begin{frame} \title{Glivenko-Translation} \begin{frame}{Glivenko-Translation} \begin{definition} \begin{itemize} \item $A^{glv} = \dneg A$ \item Kuroda-axiom $\; \kuroda A$ \end{itemize} \end{definition} Assuming Kuroda-axiom, $glv$ becomes equivalent to $kur$,$kol$ and $ggr$. \begin{theorem} $\KIbox \oplus Z \vdash \kuroda A \; \Rightarrow$ \\ $\adequate Z t$ \begin{itemize} \item [1] Assuming Kuroda-axiom, $glv$ becomes equivalent to $kur$,$kol$ and $ggr$. \item [2] $\KIbox \oplus Z \vdash \kuroda A \; \Rightarrow$\\$(\adequate Z t)$ \end{itemize} \end{theorem} \begin{Proof} Since Kuroda-axiom is a Kuroda-envelope\ldots \begin{itemize} \item [1] by straightforward induction \item [2] since Kuroda-axiom is a Kuroda-envelope\ldots \end{itemize} \end{Proof} \end{frame} ... ...
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!