fix notation in calculation of ends

[topbloke-formulae.git] / lemmas.tex

diff --git a/lemmas.tex b/lemmas.tex
index 577f5135c408bf081ccea472f6e02fa9779cd6a2..41ea1c815af7fa6e04a2237255e5b355860888ac 100644 (file)
--- a/lemmas.tex
+++ b/lemmas.tex
@@ -53,8 +53,8 @@ So by Base Acyclic $D \isin B \implies D \notin \py$.
   \end{cases}
 }\]
 
-\subsection{Tip Self Inpatch}
-Given Exclusive Tip Contents and Base Acyclic for $C$,
+\subsection{Tip Own Contents}
+Given Base Acyclic for $C$,
 $$
   \bigforall_{C \in \py} C \haspatch \p
 $$
@@ -63,7 +63,9 @@ Ie, tip commits contain their own patch.
 \proof{
 Apply Exclusive Tip Contents to some $D \in \py$:
 $ \bigforall_{C \in \py}\bigforall_{D \in \py}
-  D \isin C \equiv D \le C $
+  D \isin C \equiv D \le C $.
+Thus $C \zhaspatch \p$.
+And we can set $F=C$ giving $F \in \py \land F \le C$, so $C \haspatch \p$.
 }
 
 \subsection{Exact Ancestors}
@@ -101,13 +103,13 @@ $$
   \bigforall_{C \hasparents \set A}
     \pendsof{C}{\set P} =
       \begin{cases}
-       C \in \p : & \{ C \}
+       C \in \set P : & \{ C \}
       \\
-       C \not\in \p : & \displaystyle
+       C \not\in \set P : & \displaystyle
        \left\{ E \Big|
            \Bigl[ \Largeexists_{A \in \set A}
                        E \in \pendsof{A}{\set P} \Bigr] \land
-           \Bigl[ \Largenexists_{B \in \set A, F \in \pendsof{B}{\p}}
+           \Bigl[ \Largenexists_{B \in \set A, F \in \pendsof{B}{\set P}}
                        E \neq F \land E \le F \Bigr]
        \right\}
       \end{cases}
@@ -115,7 +117,7 @@ $$
 \proof{
 Trivial for $C \in \set P$.  For $C \not\in \set P$,
 $\pancsof{C}{\set P} = \bigcup_{A \in \set A} \pancsof{A}{\set P}$.
-So $\pendsof{C}{\set P} \subset \bigcup_{E in \set E} \pendsof{E}{\set P}$.
+So $\pendsof{C}{\set P} \subset \bigcup_{E \in \set E} \pendsof{E}{\set P}$.
 Consider some $E \in \pendsof{A}{\set P}$.  If $\exists_{B,F}$ as
 specified, then either $F$ is going to be in our result and
 disqualifies $E$, or there is some other $F'$ (or, eventually,
@@ -124,6 +126,7 @@ Otherwise, $E$ meets all the conditions for $\pends$.
 }
 
 \subsection{Ingredients Prevent Replay}
+Given conformant commits $A \in \set A$,
 $$
   \left[
     {C \hasparents \set A} \land
@@ -140,12 +143,13 @@ $$
 $$
 \proof{
   Trivial for $D = C$.  Consider some $D \neq C$, $D \isin C$.
-  By the preconditions, there is some $A$ s.t. $D \in \set A$
+  By the preconditions, there is some $A$ s.t. $A \in \set A$
   and $D \isin A$.  By No Replay for $A$, $D \le A$.  And
   $A \le C$ so $D \le C$.
 }
 
 \subsection{Simple Foreign Inclusion}
+Given a conformant commit $L$,
 $$
   \left[
     C \hasparents \{ L \}
@@ -167,6 +171,7 @@ So $D \isin C \equiv D \le C$.
 }
 
 \subsection{Totally Foreign Contents}
+Given conformant commits $A \in \set A$,
 $$
    \left[
     C \hasparents \set A \land