MyWikiBiz

Changes

Directory:Jon Awbrey/Papers/Syntactic Transformations (view source)

Revision as of 23:20, 10 February 2009

5,189 bytes added ,  23:20, 10 February 2009
→‎Derived Equivalence Relations: format fact box
Line 2,028: Line 2,028:  
<br>
 
<br>
   −
<pre>
+
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
−
Fact 2.1
+
|
−
 
+
{| align="center" cellpadding="0" cellspacing="0" width="100%"
−
If R c OxSxI,
+
|- style="height:50px; text-align:center"
−
 
+
| style="width:80%" | &nbsp;
−
then the following are identical subsets of SxI:
+
| style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~2.1}</math>
−
 
+
|}
−
F2.1a. DerR :D13a
+
|-
−
::
+
|
−
F2.1b. Der(R) :D13b
+
{| align="center" cellpadding="0" cellspacing="0" width="100%"
−
::
+
|- style="height:50px"
−
F2.1c. {<x, y> C SxI :
+
| width="2%"  style="border-top:1px solid black" | &nbsp;
−
Den(R, x) = Den(R, y)
+
| width="10%" style="border-top:1px solid black" | <math>\text{If}\!</math>
−
} :D13c
+
| width="68%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math>
−
:R9a
+
| width="20%" style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
−
::
+
|- style="height:50px"
−
F2.1d. {<x, y> C SxI :
+
| &nbsp;
−
{Den(R, x)} = {Den(R, y)}
+
| <math>\text{then}\!</math>
−
} :R9b
+
| <math>\text{the following are identical subsets of}~ S \times I :</math>
−
::
+
| style="border-left:1px solid black" | &nbsp;
−
F2.1e. {<x, y> C SxI :
+
|}
−
for all o C O
+
|-
−
{Den(R, x)}(o) = {Den(R, y)}(o)
+
|
−
} :R9c
+
{| align="center" cellpadding="0" cellspacing="0" width="100%"
−
::
+
|- style="height:60px"
−
F2.1f. {<x, y> C SxI :
+
| width="2%"  style="border-top:1px solid black" | &nbsp;
−
Conj(o C O)
+
| width="10%" style="border-top:1px solid black" | <math>\operatorname{F2.1a.}</math>
−
{Den(R, x)}(o) = {Den(R, y)}(o)
+
| width="68%" style="border-top:1px solid black" | <math>\operatorname{Der}^L</math>
−
} :R9d
+
| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{F2.1a~:~D13a}</math>
−
::
+
|- style="height:20px"
−
F2.1g. {<x, y> C SxI :
+
| &nbsp;
−
Conj(o C O)
+
| &nbsp;
−
(( {Den(R, x)}(o) , {Den(R, y)}(o) ))
+
| &nbsp;
−
} :R9e
+
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
−
::
+
|- style="height:40px"
−
F2.1h. {<x, y> C SxI :
+
| &nbsp;
−
Conj(o C O)
+
| valign="top" | <math>\operatorname{F2.1b.}</math>
−
(( {Den(R, x)} , {Den(R, y)} ))$(o)
+
| valign="top" | <math>\operatorname{Der}(L)</math>
−
} :R9f
+
| style="border-left:1px solid black; text-align:center" |
−
:D12e
+
<math>\operatorname{F2.1b~:~D13b}</math>
−
::
+
|- style="height:20px"
−
F2.1i. {<x, y> C SxI :
+
| &nbsp;
−
Conj(o C O)
+
| &nbsp;
−
(( {ROS.x} , {ROS.y} ))$(o)
+
| &nbsp;
−
} :D12a
+
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
−
</pre>
+
|- style="height:60px"
 +
| &nbsp;
 +
| valign="top" | <math>\operatorname{F2.1c.}</math>
 +
| valign="top" |
 +
<math>\begin{array}{ll}
 +
\{ & (x, y) \in S \times I ~: \\
 +
  & \operatorname{Den}(L, x) ~=~ \operatorname{Den}(L, y) \\
 +
\} & \\
 +
\end{array}</math>
 +
| style="border-left:1px solid black; text-align:center" |
 +
<p><math>\operatorname{F2.1c~:~D13c}</math></p>
 +
<p><math>\operatorname{F2.1c~:~R9a}</math></p>
 +
|- style="height:20px"
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 +
|- style="height:60px"
 +
| &nbsp;
 +
| valign="top" | <math>\operatorname{F2.1d.}</math>
 +
| valign="top" |
 +
<math>\begin{array}{ll}
 +
\{ & (x, y) \in S \times I ~: \\
 +
  & \upharpoonleft \operatorname{Den}(L, x) \upharpoonright ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright \\
 +
\} & \\
 +
\end{array}</math>
 +
| style="border-left:1px solid black; text-align:center" |
 +
<math>\operatorname{F2.1d~:~R9b}</math>
 +
|- style="height:20px"
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 +
|- style="height:60px"
 +
| &nbsp;
 +
| valign="top" | <math>\operatorname{F2.1e.}</math>
 +
| valign="top" |
 +
<math>\begin{array}{ll}
 +
\{ & (x, y) \in S \times I ~: \\
 +
  & \overset{O}{\underset{o}{\forall}}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright (o) ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright (o) \\
 +
\} & \\
 +
\end{array}</math>
 +
| style="border-left:1px solid black; text-align:center" |
 +
<math>\operatorname{F2.1e~:~R9c}</math>
 +
|- style="height:20px"
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 +
|- style="height:60px"
 +
| &nbsp;
 +
| valign="top" | <math>\operatorname{F2.1f.}</math>
 +
| valign="top" |
 +
<math>\begin{array}{ll}
 +
\{ & (x, y) \in S \times I ~: \\
 +
  & \underset{o \in O}{\operatorname{Conj}}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright (o) ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright (o) \\
 +
\} & \\
 +
\end{array}</math>
 +
| style="border-left:1px solid black; text-align:center" |
 +
<math>\operatorname{F2.1f~:~R9d}</math>
 +
|- style="height:20px"
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 +
|- style="height:60px"
 +
| &nbsp;
 +
| valign="top" | <math>\operatorname{F2.1g.}</math>
 +
| valign="top" |
 +
<math>\begin{array}{ll}
 +
\{ & (x, y) \in S \times I ~: \\
 +
  & \underset{o \in O}{\operatorname{Conj}}~ \underline{((}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright (o) ~,~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright (o) ~\underline{))} \\
 +
\} & \\
 +
\end{array}</math>
 +
| style="border-left:1px solid black; text-align:center" |
 +
<math>\operatorname{F2.1g~:~R9e}</math>
 +
|- style="height:20px"
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 +
|- style="height:60px"
 +
| &nbsp;
 +
| valign="top" | <math>\operatorname{F2.1h.}</math>
 +
| valign="top" |
 +
<math>\begin{array}{ll}
 +
\{ & (x, y) \in S \times I ~: \\
 +
  & \underset{o \in O}{\operatorname{Conj}}~ \underline{((}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright ~,~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright ~\underline{))}^\$ (o) \\
 +
\} & \\
 +
\end{array}</math>
 +
| style="border-left:1px solid black; text-align:center" |
 +
<p><math>\operatorname{F2.1h~:~R9f}</math></p>
 +
<p><math>\operatorname{F2.1h~:~D12e}</math></p>
 +
|- style="height:20px"
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 +
|- style="height:60px"
 +
| &nbsp;
 +
| valign="top" | <math>\operatorname{F2.1i.}</math>
 +
| valign="top" |
 +
<math>\begin{array}{ll}
 +
\{ & (x, y) \in S \times I ~: \\
 +
  & \underset{o \in O}{\operatorname{Conj}}~ \underline{((}~ \upharpoonleft L_{OS} \cdot x \upharpoonright ~,~ \upharpoonleft L_{OS} \cdot y \upharpoonright ~\underline{))}^\$ (o) \\
 +
\} & \\
 +
\end{array}</math>
 +
| style="border-left:1px solid black; text-align:center" |
 +
<math>\operatorname{F2.1i~:~D12a}</math>
 +
|}
 +
|}
    
<br>
 
<br>
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/78730"