From 68716d1ab46dee3dfc1b03089f941dbb6883cdcd Mon Sep 17 00:00:00 2001
From: Simon-Kor <52245124+Simon-Kor@users.noreply.github.com>
Date: Wed, 4 Sep 2024 15:17:50 +0200
Subject: Mismatched Assume in Induction

Unexpeced Mismatch in line 99 and 109  or the pair
100 and 108. Parsing works with line 99 and 108 or with
the lines 100 and 109.

zf: MismatchedAssume (TermSymbol (SymbolPredicate (PredicateNoun (SgPl {sg = [Just (Word "natural"),Just (Word "number")], pl = [Just (Word "natural"),Just (Word "numbers")]}))) [TermVar (NamedVar "n")]) (Connected Implication (TermSymbol (SymbolPredicate (PredicateRelation (Command "in"))) [TermVar (NamedVar "n"),TermSymbol (SymbolMixfix [Just (Command "naturals")]) []]) (TermSymbol (SymbolPredicate (PredicateVerb (SgPl {sg = [Just (Word "has"),Just (Word "cardinality"),Nothing], pl = [Just (Word "ha"),Just (Word "cardinality"),Nothing]}))) [TermSymbol (SymbolMixfix [Just (Command "seq"),Just InvisibleBraceL,Nothing,Just InvisibleBraceR,Just InvisibleBraceL,Nothing,Just InvisibleBraceR]) [TermSymbol (SymbolMixfix [Just (Command "emptyset")]) [],TermVar (NamedVar "n")],TermSymbol (SymbolMixfix [Just (Command "suc"),Just InvisibleBraceL,Nothing,Just InvisibleBraceR]) [TermVar (NamedVar "n")]]))
---
 library/topology/urysohn2.tex | 108 +++++++++++++++++++++++-------------------
 1 file changed, 59 insertions(+), 49 deletions(-)

(limited to 'library/topology/urysohn2.tex')

diff --git a/library/topology/urysohn2.tex b/library/topology/urysohn2.tex
index dbcfc53..a3f3ea4 100644
--- a/library/topology/urysohn2.tex
+++ b/library/topology/urysohn2.tex
@@ -12,32 +12,11 @@
 \import{set/powerset.tex}
 \import{set/fixpoint.tex}
 \import{set/product.tex}
+\import{topology/real-topological-space.tex}
 
 \section{Urysohns Lemma}
 
-\begin{definition}\label{minimum}
-    $\min{X} = \{x \in X \mid \forall y \in X. x \leq y \}$.
-\end{definition}
-
-
-\begin{definition}\label{maximum}
-    $\max{X} = \{x \in X \mid \forall y \in X. x \geq y \}$.
-\end{definition}
-
-
-\begin{definition}\label{intervalclosed}
-    $\intervalclosed{a}{b} = \{x \in \reals \mid a \leq x \leq b\}$.
-\end{definition}
-
-
-\begin{definition}\label{intervalopen}
-    $\intervalopen{a}{b} = \{ x \in \reals \mid a < x < b\}$.
-\end{definition}
-
 
-\begin{definition}\label{one_to_n_set}
-    $\seq{m}{n} = \{x \in \naturals \mid  m \leq x \leq n\}$.   
-\end{definition}
 
 
 \begin{definition}\label{sequence}
@@ -89,26 +68,6 @@
 \end{definition}
 
 
-\begin{axiom}\label{abs_behavior1}
-    If $x \geq \zero$ then $\abs{x} = x$.
-\end{axiom}
-
-\begin{axiom}\label{abs_behavior2}
-    If $x < \zero$ then $\abs{x} = \neg{x}$.
-\end{axiom}
-
-\begin{definition}\label{realsminus}
-    $\realsminus = \{r \in \reals \mid r < \zero\}$.
-\end{definition}
-
-\begin{definition}\label{realsplus}
-    $\realsplus = \reals \setminus \realsminus$.
-\end{definition}
-
-\begin{definition}\label{epsilon_ball}
-    $\epsBall{x}{\epsilon} = \intervalopen{x-\epsilon}{x+\epsilon}$.
-\end{definition}
-
 \begin{definition}\label{pointwise_convergence}
     $X$ converge to $x$ iff for all $\epsilon \in \realsplus$ there exist $N \in \dom{X}$ such that for all $n \in \dom{X}$ such that $n > N$ we have $\at{X}{n} \in \epsBall{x}{\epsilon}$.
 \end{definition}
@@ -120,10 +79,61 @@
     Then $X$ is a sequence.
 \end{proposition}
 
-\begin{definition}\label{higher_urysohn_chain}
+\begin{definition}\label{lifted_urysohn_chain}
     $U$ is a lifted urysohnchain of $X$ iff $U$ is a sequence and $\dom{U} = \naturals$ and for all $n \in \dom{U}$ we have $\at{U}{n}$ is a urysohnchain of $X$ and $\at{U}{\suc{n}}$ is a minimal finer extention of $\at{U}{n}$ in $X$.
 \end{definition}
 
+\begin{definition}\label{normal_ordered_urysohnchain}
+    $U$ is normal ordered iff there exist $n \in \naturals$ such that $\dom{U} = \seq{\zero}{n}$.
+\end{definition}
+
+\begin{definition}\label{bijection_of_urysohnchains}
+    $f$ is consistent on $X$ to $Y$ iff $f$ is a bijection from $\dom{X}$ to $\dom{Y}$ and for all $n,m \in \dom{X}$ such that $n < m$ we have $f(n) < f(m)$.
+\end{definition}
+
+\begin{proposition}\label{naturals_in_transitive}
+    $\naturals$ is a \in-transitive set.
+\end{proposition}
+
+\begin{proposition}\label{naturals_elem_in_transitive}
+    If $n \in \naturals$ then $n$ is \in-transitive and every element of $n$ is \in-transitive. 
+    %If $n$ is a natural number then $n$ is \in-transitive and every element of $n$ is \in-transitive. 
+    
+\end{proposition}
+
+\begin{proposition}\label{seq_zero_to_n_isomorph_to_n}
+    For all $n \in \naturals$ we have $\seq{\zero}{n}$ has cardinality $\suc{n}$.
+\end{proposition}
+\begin{proof} [Proof by \in-induction on $n$]
+    Assume $n \in \naturals$.
+    %Assume $n$ is a natural number.
+    %We show that $\seq{\zero}{\zero}$ has cardinality $1$.
+    %\begin{subproof}
+    %    It suffices to show that $1 = \seq{\zero}{\zero}$.
+    %    Follows by set extensionality.
+    %\end{subproof}
+    %For all $n \in \naturals$ we have either $n = \zero$ or there exist $m \in \naturals$ such that $n = \suc{m}$.
+    %For all $n \in \naturals$ we have either $n = \zero$ or $\zero \in n$.
+    %We show that if $\seq{\zero}{n}$ has cardinality $\suc{n}$ then $\seq{\zero}{\suc{n}}$ has cardinality $\suc{\suc{n}}$.
+    %\begin{subproof}
+    %    Omitted.
+    %\end{subproof}
+\end{proof}
+
+\begin{proposition}\label{existence_normal_ordered_urysohn}
+    Let $X$ be a urysohn space.
+    Suppose $U$ is a urysohnchain of $X$.
+    Suppose $\dom{U}$ is finite.
+    Then there exist $V,f$ such that $V$ is a urysohnchain of $X$ and $f$ is consistent on $X$ to $Y$ and $V$ is normal ordered.
+\end{proposition}
+\begin{proof}
+    Take $k$ such that $\dom{U}$ has cardinality $k$ by \cref{ran_converse,cardinality,finite}.
+    There exist $F$ such that $F$ is a bijection from $\seq{\zero}{k}$ to $\dom{U}$.
+
+
+\end{proof}
+
+
 \begin{definition}\label{staircase}
     $f$ is a staircase function adapted to $U$ in $X$ iff $U$ is a urysohnchain of $X$ and for all $x,n,m,k$ such that $k = \max{\dom{U}}$ and $n,m \in \dom{U}$ and $n$ follows $m$ in $\dom{U}$ and $x \in (\at{U}{m} \setminus \at{U}{n})$ we have $f(x)= \rfrac{m}{k}$.
 \end{definition}
@@ -136,9 +146,6 @@
 
 
 
-
-
-
 \begin{theorem}\label{urysohnsetinbeetween}
     Let $X$ be a urysohn space.
     Suppose $A,B \in \closeds{X}$.
@@ -159,9 +166,9 @@
     There exist $U$ such that $U$ is a sequence and $\dom{U} = \naturals$ and $\at{U}{\zero} = U_0$ and for all $n \in \dom{U}$ we have $\at{U}{n}$ is a urysohnchain of $X$ and $\at{U}{\suc{n}}$ is a minimal finer extention of $\at{U}{n}$ in $X$.
 \end{theorem}
 \begin{proof}
-    $U_0$ is a urysohnchain of $X$.
-    It suffices to show that for all $V$ such that $V$ is a urysohnchain of $X$ there exist $V'$ such that $V'$ is a urysohnchain of $X$ and $V'$ is a minimal finer extention of $V$ in $X$.
-    
+    %$U_0$ is a urysohnchain of $X$.
+    %It suffices to show that for all $V$ such that $V$ is a urysohnchain of $X$ there exist $V'$ such that $V'$ is a urysohnchain of $X$ and $V'$ is a minimal finer extention of $V$ in $X$.
+    Omitted.
 \end{proof}
 
 
@@ -246,6 +253,9 @@
     \begin{subproof}
         Follows by \cref{chain_of_subsets,urysohnchain,induction_on_urysohnchains}.
     \end{subproof}
+    Take $U$ such that $U$ is a lifted urysohnchain of $X$ and $\at{U}{\zero} = U_0$.
+
+
 
 
 
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