diff options
| -rw-r--r-- | .gitignore | 4 | ||||
| -rw-r--r-- | library/topology/basis.tex | 20 | ||||
| -rw-r--r-- | vampire-taks-with-unexpacted-behavoir/50-plus-secounds-for-iff-stamtent.p | 6 | ||||
| -rw-r--r-- | vampire-taks-with-unexpacted-behavoir/topological-basis-behovoir.p | 26 |
4 files changed, 45 insertions, 11 deletions
@@ -41,6 +41,6 @@ premseldump/ haddocks/ stack.yaml.lock zf*.svg -Anmerkungen.txt -vampire-taks-with-not-expacted-behavoir/ + + diff --git a/library/topology/basis.tex b/library/topology/basis.tex index 61a358f..6fc07d3 100644 --- a/library/topology/basis.tex +++ b/library/topology/basis.tex @@ -79,36 +79,38 @@ \begin{lemma}\label{inters_in_genopens} Assume $B$ is a topological basis for $X$. - %For all $A, C$ - If $A\in \genOpens{B}{X}$ and $C\in \genOpens{B}{X}$ then $(A\inter C) \in \genOpens{B}{X}$. + Suppose $A, C\in \genOpens{B}{X}$. + + Then $(A\inter C) \in \genOpens{B}{X}$. \end{lemma} \begin{proof} Show $(A \inter C) \in \pow{X}$. \begin{subproof} - $(A \inter C) \subseteq X$ by assumption. + Omitted. \end{subproof} - Therefore for all $A, C \in \genOpens{B}{X}$ we have $(A \inter C) \in \pow{X}$. Show for all $x\in (A\inter C)$ there exists $W \in B$ such that $x\in W$ and $W \subseteq (A\inter C)$. \begin{subproof} Fix $x \in (A\inter C)$. - There exist $V' \in B$ such that $x \in V'$ and $V' \subseteq A$ by assumption. %TODO: Warum muss hier by assumtion hin? - There exist $V'' \in B$ such that $x \in V''$ and $V'' \subseteq C$ by assumption. - There exist $W \in B$ such that $x \in W$ and $W \subseteq v'$ and $W \subseteq V''$ by assumption. + $x \in A,C$. + There exist $V' \in B$ such that $x \in V'$ and $V' \subseteq A$ by \cref{genopens}. + There exist $V'' \in B$ such that $x \in V''$ and $V'' \subseteq C$ by \cref{genopens}. + $x \in (V' \inter V'')$. + There exist $W \in B$ such that $x \in W$ and $W \subseteq V'$ and $W \subseteq V''$. Show $W \subseteq (A\inter C)$. \begin{subproof} %$W \subseteq v'$ and $W \subseteq V''$. - For all $y \in W$ we have $y \in V'$ and $y \in V''$ by assumption. + For all $y \in W$ we have $y \in V'$ and $y \in V''$. \end{subproof} \end{subproof} %Therefore for all $A, C, x$ such that $A \in \genOpens{B}{X}$ and $C \in \genOpens{B}{X}$ and $x \in (A \inter C)$ we have there exists $W \in B$ %such that $x\in W$ and $W \subseteq (A\inter C)$. - $(A\inter C) \in \genOpens{B}{X}$ by assumption. + $(A\inter C) \in \genOpens{B}{X}$. \end{proof} diff --git a/vampire-taks-with-unexpacted-behavoir/50-plus-secounds-for-iff-stamtent.p b/vampire-taks-with-unexpacted-behavoir/50-plus-secounds-for-iff-stamtent.p new file mode 100644 index 0000000..19ed046 --- /dev/null +++ b/vampire-taks-with-unexpacted-behavoir/50-plus-secounds-for-iff-stamtent.p @@ -0,0 +1,6 @@ +fof(inters_in_genopens,conjecture,unions(fB)=fX). +fof(topological_basis,axiom,![XB,XX]:(topological_basis(XB,XX)<=>(unions(XB)=XX&![XU,XV,Xx]:((elem(XU,XB)&elem(XV,XB)&elem(Xx,XU)&elem(Xx,XV))=>?[XW]:(elem(XW,XB)&elem(Xx,XW)&subseteq(XW,XU)&subseteq(XW,XV)))))). +fof(inters_in_genopens1,axiom,elem(fx,fVprimeprime)&subseteq(fVprimeprime,fC)&elem(fVprimeprime,fB)). +fof(inters_in_genopens2,axiom,elem(fx,fVprime)&subseteq(fVprime,fA)&elem(fVprime,fB)). +fof(inters_in_genopens3,axiom,elem(fx,inter(fA,fC))). +fof(inters_in_genopens4,axiom,topological_basis(fB,fX)). diff --git a/vampire-taks-with-unexpacted-behavoir/topological-basis-behovoir.p b/vampire-taks-with-unexpacted-behavoir/topological-basis-behovoir.p new file mode 100644 index 0000000..76b03e4 --- /dev/null +++ b/vampire-taks-with-unexpacted-behavoir/topological-basis-behovoir.p @@ -0,0 +1,26 @@ +% It doesn't make sense for me that this task takes so long. It taks on my computer nearly 50-60 secounds. + +fof(inters_in_genopens,conjecture,?[XW]:(elem(XW,fB)&elem(fx,XW)&subseteq(XW,fvprime)&subseteq(XW,fVprimeprime))). +%fof(topological_basis,axiom,![XB,XX]:(topological_basis(XB,XX)<=>(unions(XB)=XX&![XU,XV,Xx]:((elem(XU,XB)&elem(XV,XB)&elem(Xx,XU)&elem(Xx,XV))=>?[XW]:(elem(XW,XB)&elem(Xx,XW)&subseteq(XW,XU)&subseteq(XW,XV)))))). +fof(inters_in_genopens1,axiom,elem(fx,fVprimeprime)&subseteq(fVprimeprime,fC)&elem(fVprimeprime,fB)). +fof(inters_in_genopens2,axiom,elem(fx,fVprime)&subseteq(fVprime,fA)&elem(fVprime,fB)). +fof(inters_in_genopens3,axiom,elem(fx,inter(fA,fC))). +fof(inters_in_genopens4,axiom,topological_basis(fB,fX)). + +fof(topological_basis1,axiom,![XU,XV,Xx]:((elem(XU,fB)&elem(XV,fB)&elem(Xx,XU)&elem(Xx,XV))=>?[XW]:(elem(XW,fB)&elem(Xx,XW)&subseteq(XW,XU)&subseteq(XW,XV)))). + +fof(inters_in_genopens10,axiom,subseteq(fVprime,fA)). +fof(inters_in_genopens11,axiom,subseteq(fVprimeprime,fC)). +fof(inters_in_genopens12,axiom,elem(fx,fVprime)). +fof(inters_in_genopens13,axiom,elem(fx,fVprimeprime)). + + +% the task (unions(fB)=fX) takes only 0.2 secounds +% but if it is stated as an axiom it doesn't effect any proof time. + + +% with this axiom it takes 27 secounds so it improve the overall time, +% but this axiom is just a "and" more and combines axiom 1 and 2 +% fof(inters_in_genopens5,axiom,elem(fVprimeprime,fB)&elem(fVprimeprime,fB)&elem(fx,fVprime)&elem(fx,fVprimeprime)). + +% fof(topological_basis1,axiom,![XU,XV,Xx]:((elem(XU,fB)&elem(XV,fB)&elem(Xx,XU)&elem(Xx,XV))=>?[XW]:(elem(XW,fB)&elem(Xx,XW)&subseteq(XW,XU)&subseteq(XW,XV)))).
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