Commit 78a8fbfc by Miriam

### merge presentation parts

parent aeae78e0
 ... ... @@ -7,6 +7,7 @@ \usepackage{tikz} \usepackage{amsmath} \usepackage{url} \usepackage{bussproofs} \usepackage[utf8]{inputenc} \renewcommand{\baselinestretch}{.99} ... ... @@ -43,7 +44,7 @@ \vspace{2cm} Now we need a proof system satisfying: Now we need a proof system satisfying exactly: \begin{itemize} \item All intuitionistic tautologies \item Closure under substitution and MP ... ... @@ -60,8 +61,58 @@ %\end{frame} \begin{frame}{Natural Deduction for $\KIbox \oplus Z$} Some of the rules \dots \scriptsize %permutations \begin{prooftree} \AxiomC{$\KIbox \oplus Z \vdash A, G \Rightarrow A$} \LeftLabel{\scriptsize(Perm)} \RightLabel{\scriptsize $G'$ is permutation of $G$} \UnaryInfC{$\KIbox \oplus Z \vdash A, G' \Rightarrow A$} \end{prooftree} %In \begin{prooftree} \AxiomC{} \LeftLabel{\scriptsize(In)} \UnaryInfC{$\KIbox \oplus Z \vdash A, G \Rightarrow A$} \end{prooftree} %Ax \begin{prooftree} \AxiomC{} \LeftLabel{\scriptsize(Ax)} \RightLabel{\scriptsize s substitution} \UnaryInfC{$\KIbox \oplus Z \vdash A, G \Rightarrow s(Z)$} \end{prooftree} %or elimination % KIbox Z G (f1 ||| f2) -> KIbox Z (f1 :: G) f3 -> KIbox Z (f2 :: G) f3 -> KIbox Z G f3 \begin{prooftree} \AxiomC{$\KIbox \oplus Z \vdash G \Rightarrow A \lor B$} \AxiomC{$\KIbox \oplus Z \vdash A, G \Rightarrow C$} \AxiomC{$\KIbox \oplus Z \vdash B, G \Rightarrow C$} \LeftLabel{\scriptsize$(\lor_E)$} \TrinaryInfC{$\KIbox \oplus Z \vdash G \Rightarrow C$} \end{prooftree} \end{frame} \begin{frame}{Natural Deduction for $\KIbox \oplus Z$} The only rule for box: %(forall f, In f F -> KIbox Z G (|[]| f)) -> KIbox Z F h -> KIbox Z G (|[]| h) \begin{prooftree} \AxiomC{$\forall A_i. \KIbox \oplus Z \vdash G \Rightarrow \Box A_i$} \AxiomC{$\KIbox \oplus Z \vdash A_1 \dots A_n \Rightarrow B$} \LeftLabel{\scriptsize($\Box_{IE}$)} \BinaryInfC{$\KIbox \oplus Z \vdash G \Rightarrow \Box B$} \end{prooftree} TODO explanaition? \end{frame} \begin{frame} IM Tautologien (impl, and, or) ... ...
 ... ... @@ -69,7 +69,7 @@ Fixpoint sat (m : kripke_model) (w : worlds _) (f : form) : Prop := | var n => valuation _ w n | A ->> B => forall w', (int_relation (_ m) w w') -> sat m w' A -> sat m w' B | A & B => sat m w A /\ sat m w B | A \v/ B => sat m w A \/ sat m w B | A ||| B => sat m w A \/ sat m w B | |[]| A => forall w', (modal_relation (_ m) w w') -> sat m w' A end . ... ... @@ -92,13 +92,32 @@ Qed. (** ** Examples *) Lemma and_imp_neg_sem: Lemma imp_dneg_sem: forall A, val (A ->> -+! A). Proof. red. intros. simpl. intuition. eapply H2. eapply preord_refl. apply int_relation_preorder. eapply upclosed_int. apply H0. apply H1. Qed. Inductive one_point_set : Set := Elem. Definition empty_relation A : relation A := (fun a => fun b => False). Program Definition one_point_frame : kripke_frame. apply (Build_kripke_frame one_point_set (empty_relation one_point_set) (empty_relation one_point_set)). - constructor. apply Elem. - Lemma dneg_imp_sem: forall A, ~ val (-+! A ->> A). Proof. unfold val. intros A H. assert (exists m w, ~sat m w (-+! A ->> A)). (** ** Soundness of KIbox *) ... ...
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