From 6839f79e775d8ffa36d421f4e3c08528323eaf17 Mon Sep 17 00:00:00 2001
From: Simon-Kor <52245124+Simon-Kor@users.noreply.github.com>
Date: Thu, 20 Jun 2024 12:59:41 +0200
Subject: [Report] Task failed and was proven in the same time.

The Line 182 has the goal to apply regular in a topological space. But if i try to prove it with Naproche ZF, then the task will be proven and the task fails in the same time. Console Output is the following:

time stack exec zf -- --log library/topology/separation.tex -t 30  --dump ./dump
[Info] 00:00.11 fof(eqvilance_teethree_closed_neighbourhood_in_open,conjecture,elem(fx,fB)&subseteq(fB,setminus(s__carrier(fX),fA))&subseteq(setminus(s__carrier(fX),fA),fU)).
[Info] 00:00.11 fof(eqvilance_teethree_closed_neighbourhood_in_open,conjecture,setminus(s__carrier(fX),setminus(s__carrier(fX),fU))=fU).
[Info] 00:00.11 fof(eqvilance_teethree_closed_neighbourhood_in_open,conjecture,?[XN]:(elem(XN,neighbourhoods(fx,fX))&subseteq(XN,fU)&is_closed_in(XN,fX))).
[Info] 00:00.21 fof(eqvilance_teethree_closed_neighbourhood_in_open,conjecture,(is_regular(fX)&is_kolmogorov(fX))<=>![XU]:(elem(XU,s__opens(fX))=>![Xx]:(elem(Xx,XU)=>?[XN]:(elem(XN,neighbourhoods(Xx,fX))&subseteq(XN,XU)&is_closed_in(XN,fX))))).
[Info] 00:02.34 fof(eqvilance_teethree_closed_neighbourhood_in_open,conjecture,?[XB,XA]:(elem(fx,XB)&subseteq(fC,XA)&inter(XA,XB)=emptyset&elem(XA,s__opens(fX))&elem(XB,s__opens(fX)))).
[Info] 00:02.55 fof(eqvilance_teethree_closed_neighbourhood_in_open,conjecture,subseteq(fB,setminus(s__carrier(fX),fA))).
[Info] 00:08.11 fof(eqvilance_teethree_closed_neighbourhood_in_open,conjecture,elem(fN,closeds(fX))&elem(fx,fN)&subseteq(fN,fU)).
[Info] 00:10.70 fof(eqvilance_teethree_closed_neighbourhood_in_open,conjecture,subseteq(setminus(s__carrier(fX),fA),setminus(s__carrier(fX),setminus(s__carrier(fX),fU)))).
[Info] 00:10.98 fof(eqvilance_teethree_closed_neighbourhood_in_open,conjecture,elem(fN,neighbourhoods(fx,fX))).
Verification failed: prover found countermodel
fof(eqvilance_teethree_closed_neighbourhood_in_open,conjecture,?[XB,XA]:(elem(fx,XB)&subseteq(fC,XA)&inter(XA,XB)=emptyset&elem(XA,s__opens(fX))&elem(XB,s__opens(fX)))).fof(is_regular,axiom,![XX]:(is_regular(XX)<=>![XV]:![Xp,XC]:((elem(Xp,s__carrier(XX))&~elem(Xp,XC)&elem(XC,closeds(XX)))=>?[XU,XC]:(elem(XU,s__opens(XX))&elem(XC,s__opens(XX))&elem(Xp,XU)&subseteq(XC,XV)&inter(XU,XV)=emptyset)))).
fof(eqvilance_teethree_closed_neighbourhood_in_open1,axiom,elem(fx,s__carrier(fX))).
fof(eqvilance_teethree_closed_neighbourhood_in_open2,axiom,~elem(fx,fC)).
fof(eqvilance_teethree_closed_neighbourhood_in_open3,axiom,elem(fC,closeds(fX))).
fof(eqvilance_teethree_closed_neighbourhood_in_open4,axiom,fC=setminus(s__carrier(fX),fU)).
fof(eqvilance_teethree_closed_neighbourhood_in_open5,axiom,elem(fx,fU)).
fof(eqvilance_teethree_closed_neighbourhood_in_open6,axiom,elem(fU,s__opens(fX))).
fof(eqvilance_teethree_closed_neighbourhood_in_open7,axiom,is_regular(fX)&is_kolmogorov(fX)).
fof(eqvilance_teethree_closed_neighbourhood_in_open8,axiom,inhabited(fX)).
fof(eqvilance_teethree_closed_neighbourhood_in_open9,axiom,topological_space(fX)).

real    0m14.914s
user    0m37.867s
sys     0m1.956s
---
 library/topology/separation.tex | 101 +++++++++++++++++++++++++++++++++-------
 1 file changed, 85 insertions(+), 16 deletions(-)

(limited to 'library/topology/separation.tex')

diff --git a/library/topology/separation.tex b/library/topology/separation.tex
index 3268bb9..c53b6a8 100644
--- a/library/topology/separation.tex
+++ b/library/topology/separation.tex
@@ -1,5 +1,5 @@
 \import{topology/topological-space.tex}
-
+\import{set.tex}
 
 % T0 separation
 \begin{definition}\label{is_kolmogorov}
@@ -146,28 +146,65 @@
     \end{subproof}
 \end{proof}
 
-\begin{definition}\label{regular}
+\begin{definition}\label{is_regular}
     $X$ is regular iff for all $C,p$ such that $p \in \carrier[X]$ and $p \notin C \in \closeds{X}$ we have there exists $U,C \in \opens[X]$ such that $p \in U$ and $C \subseteq V$ and $U \inter V = \emptyset$.
 \end{definition}
 
-\begin{definition}\label{regular_space}
+\begin{abbreviation}\label{regular_space}
     $X$ is a regular space iff $X$ is a topological space and $X$ is regular.
-\end{definition}
+\end{abbreviation}
 
 
-\begin{definition}\label{teethree}
+\begin{abbreviation}\label{teethree}
     $X$ is \teethree\ iff $X$ is regular and $X$ is \teezero\ .
-\end{definition}
+\end{abbreviation}
 
-\begin{definition}\label{teethree_space}
+\begin{abbreviation}\label{teethree_space}
     $X$ is a \teethree-space iff $X$ is a topological space and $X$ is \teethree\ .
-\end{definition}
+\end{abbreviation}
+
+\begin{proposition}\label{eqvilance_teethree_closed_neighbourhood_in_open}
+    Let $X$ be a topological space.
+    Suppose $X$ is inhabited.
+    $X$ is \teethree\ iff for all $U \in \opens[X]$ we have for all $x \in U$ we have there exist $N \in \neighbourhoods{x}{X}$ such that $N \subseteq U$ and $N$ is closed in $X$.    
+\end{proposition}
+\begin{proof}
+
+    We show that if $X$ is \teethree\ then for all $U \in \opens[X]$ we have for all $x \in U$ we have there exist $N \in \neighbourhoods{x}{X}$ such that $N \subseteq U$ and $N$ is closed in $X$.    
+    \begin{subproof}
+        Suppose $X$ is regular and kolmogorov.
+        Fix $U \in \opens[X]$.
+        Fix $x \in U$.
+        Let $C = \carrier[X] \setminus U$.
+        Then $C \in \closeds{X}$.
+        $x \notin C$.
+        $x \in \carrier[X]$.
+        There exists $A,B \in \opens[X]$ such that $x \in B$ and $C \subseteq A$ and $A \inter B = \emptyset$ by \cref{is_regular}.        
+        $B \subseteq (\carrier[X] \setminus A)$.
+        $(\carrier[X] \setminus A) \subseteq (\carrier[X] \setminus (\carrier[X] \setminus U))$.
+        $(\carrier[X] \setminus (\carrier[X] \setminus U)) = U$.
+        $x \in B \subseteq (\carrier[X] \setminus A) \subseteq U$.
+        Let $N = (\carrier[X] \setminus A)$.
+        Then $N \in \closeds{X}$ and $x \in N$ and $N \subseteq U$.
+        Particularly, $N \in \neighbourhoods{x}{X}$.
+    \end{subproof}
+    We show that if for all $U \in \opens[X]$ we have for all $x \in U$ we have there exist $N \in \neighbourhoods{x}{X}$ such that $N \subseteq U$ and $N$ is closed in $X$ then $X$ is \teethree\ .    
+    \begin{subproof}
+        Omitted.
+    \end{subproof}
+\end{proof}
+
+
+
 
 \begin{proposition}\label{teethree_space_is_teetwo_space}
     Let $X$ be a \teethree-space.
+    Suppose $X$ is inhabited.
     Then $X$ is a \teetwo-space.
 \end{proposition}
 \begin{proof}
+    Omitted.
+\end{proof}
     For all $x,y \in \carrier[X]$ such that $x \neq y$ we have $x \notin \{y\}$.
     It suffices to show that $X$ is hausdorff.
     It suffices to show that for all $x \in \carrier[X]$ we have for all $y \in \carrier[X]$ such that $y \neq x$ we have there exist $U,V \in \opens[X]$ such that $x\in U$ and $y \in V$ and $U$ is disjoint from $V$.
@@ -175,16 +212,48 @@
     It suffices to show that for all $y \in \carrier[X]$ such that $y \neq x$ we have there exist $U,V \in \opens[X]$ such that $x\in U$ and $y \in V$ and $U$ is disjoint from $V$.
     Fix $y \in \carrier[X]$.
 
-    %There exist $U' \in \opens[X]$ such that $x\in U'\not\ni y$ or $x\notin U'\ni y$ by \cref{}.
-    %There exist $C \in \closeds{X}$ such that $y \in C \not\ni X$.
+
     We show that there exist $U,V,C$ such that $U,V \in \opens[X]$ and $C\in \closeds{X}$ and $x \in U$ and $y \in C \subseteq V$ and $U$ is disjoint from $V$.
     \begin{subproof}
-        Omitted.
+        There exist $C' \in \closeds{X}$ such that $x \in \carrier[X]$ and $x \notin C' \in \closeds{X}$ and there exists $U',V' \in \opens[X]$ such that $x \in U'$ and $C' \subseteq V'$ and $U' \inter V' = \emptyset$.
+        There exists $U',V' \in \opens[X]$ such that $x \in U'$ and $C' \subseteq V'$ and $U' \inter V' = \emptyset$.
+        $U'$ is disjoint from $V'$.
+        $x \in U'$.
+        $x \notin C' \subseteq V'$.
+        $U',V' \in \opens[X]$.
+        $C' \in \closeds{X}$.
+        We show that there exist $K \in \closeds{X}$ such that $x \notin K \ni y$.
+        \begin{subproof}
+            $X$ is Kolmogorov.
+            For all $x',y'\in\carrier[X]$ such that $x'\neq y'$ there exist $H\in\opens[X]$ such that $x'\in H\not\ni y'$ or $x'\notin H\ni y'$.
+            we show that there exist $H \in \opens[X]$ such that $x \notin H \ni y$ or $y \notin H \ni x$.
+            \begin{subproof}
+                Omitted.
+            \end{subproof}
+            $H \subseteq \carrier[X]$ by \cref{opens_type,subseteq}.
+            Since $\carrier[X] \ni x \notin H$ or $\carrier[X] \ni y \notin H$, we have there exist $c \in H$. 
+            Then $H \neq \carrier[X]$.
+            Since $y \in H$ or $x \in H$, we have $H \neq \emptyset$.
+            Let $K = \carrier[X] \setminus H$.
+            $K$ is inhabited.
+            $K \in \closeds{X}$ by \cref{complement_of_open_is_closed}.
+            $x \notin K \ni y$ or $y \notin K \ni x$.
+            \begin{byCase}
+                \caseOf{$y \in K$.} Trivial.
+                \caseOf{$y \notin K$.}
+                Then there exist $U'',V'' \in \opens[X]$ such that $x \in U''$ and $K \subseteq V''$ and $U'' \inter V'' = \emptyset$ by \cref{is_regular}.
+                Let $K' = \carrier[X] \setminus U''$.
+                $x \in K'$.
+                $K' \in \closeds{X}$.
+            \end{byCase}
+
+
+        \end{subproof}
+
+        Follows by assumption.
     \end{subproof}
-    $y \in V$.
+    $y \in V$ by assumption.
     Follows by assumption.
-\end{proof}
 
-%    for all $x,y\in\carrier[X]$ such that $x\neq y$
-%    there exist $U, V\in\opens[X]$ such that
-%    $x\in U$ and $y\in V$ and $U$ is disjoint from $V$.
\ No newline at end of file
+    
+%\end{proof}
-- 
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