Changes

MyWikiBiz, Author Your Legacy — Wednesday June 12, 2024
Jump to navigationJump to search

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

Revision as of 22:54, 12 February 2009

5,496 bytes added ,  22:54, 12 February 2009
→‎Derived Equivalence Relations: format fact box
Line 2,187: Line 2,187:  
<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 2.2
+
|
−
 
+
{| 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 equivalent:
+
| style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~2.2}</math>
−
 
+
|}
−
F2.2a. DerR = {<x, y> C SxI :
+
|-
−
Conj(o C O)
+
|
−
{Den(R, x)}(o) =
+
{| align="center" cellpadding="0" cellspacing="0" width="100%"
−
{Den(R, y)}(o)
+
|- style="height:50px"
−
} :R11a
+
| width="2%"  style="border-top:1px solid black" | &nbsp;
−
::
+
| width="12%" style="border-top:1px solid black" | <math>\text{If}\!</math>
−
F2.2b. {DerR} = { {<x, y> C SxI :
+
| width="66%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math>
−
Conj(o C O)
+
| width="20%" style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
−
{Den(R, x)}(o) =
+
|- style="height:50px"
−
{Den(R, y)}(o)
+
| &nbsp;
−
}
+
| <math>\text{then}\!</math>
−
} :R11b
+
| <math>\text{the following are equivalent:}\!</math>
−
::
+
| style="border-left:1px solid black" | &nbsp;
−
F2.2c. {DerR} c SxIxB
+
|}
−
:
+
|-
−
{DerR} = {<x, y, v> C SxIxB :
+
|
−
v =
+
{| align="center" cellpadding="0" cellspacing="0" width="100%"
−
[ Conj(o C O)
+
|- style="height:10px"
−
{Den(R, x)}(o) =
+
| width="2%"  style="border-top:1px solid black" | &nbsp;
−
{Den(R, y)}(o)
+
| width="12%" style="border-top:1px solid black" | &nbsp;
−
]
+
| width="66%" style="border-top:1px solid black" | &nbsp;
−
} :R11c
+
| width="20%" style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
−
::
+
|- style="height:100px"
−
F2.2d. {DerR} = {<x, y, v> C SxIxB :
+
| &nbsp;
−
v =
+
| valign="top" | <math>\operatorname{F2.2a.}</math>
−
Conj(o C O)
+
| valign="top" |
−
[ {Den(R, x)}(o) =
+
<math>\begin{array}{cccl}
−
{Den(R, y)}(o)
+
\operatorname{Der}^L
−
]
+
& = & \{ & (x, y) \in S \times I ~: \\
−
} :Log
+
&  &    & \begin{array}{ccl}
−
 
+
          \underset{o \in O}{\operatorname{Conj}} \\
−
F2.2e. {DerR} = {<x, y, v> C SxIxB :
+
          & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
−
v =
+
          & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
−
Conj(o C O)
+
          & ) & \\
−
(( {Den(R, x)}(o),
+
          \end{array} \\
−
{Den(R, y)}(o)
+
&  & \} & \\
−
))
+
\end{array}</math>
−
} :Log
+
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.2a~:~R11a}</math>
−
 
+
|- style="height:20px"
−
F2.2f. {DerR} = {<x, y, v> C SxIxB :
+
| colspan="3" | &nbsp;
−
v =
+
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
−
Conj(o C O)
+
|- style="height:100px"
−
(( {Den(R, x)},
+
| &nbsp;
−
{Den(R, y)}
+
| valign="top" | <math>\operatorname{F2.2b.}</math>
−
))$(o)
+
| valign="top" |
−
} :$
+
<math>\begin{array}{ccccl}
−
</pre>
+
\upharpoonleft \operatorname{Der}^L \upharpoonright
 +
& = & \upharpoonleft  & \{ & (x, y) \in S \times I ~: \\
 +
&  &                &    & \begin{array}{ccl}
 +
                            \underset{o \in O}{\operatorname{Conj}} \\
 +
                            & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
 +
                            & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
 +
                            & ) & \\
 +
                            \end{array} \\
 +
&  &                & \} & \\
 +
&  & \upharpoonright &    & \\
 +
\end{array}</math>
 +
| style="border-left:1px solid black; text-align:center" |
 +
<math>\operatorname{F2.2b~:~R11b}</math>
 +
|- style="height:20px"
 +
| colspan="3" | &nbsp;
 +
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 +
|- style="height:100px"
 +
| &nbsp;
 +
| valign="top" | <math>\operatorname{F2.2c.}</math>
 +
| valign="top" |
 +
<math>\begin{array}{cccl}
 +
\upharpoonleft \operatorname{Der}^L \upharpoonright
 +
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
 +
&  &    & \begin{array}{cccl}
 +
          \downharpoonleft  & \underset{o \in O}{\operatorname{Conj}} \\
 +
          &                & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
 +
          &                & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
 +
          &                & ) & \\
 +
          \downharpoonright &  & \\
 +
          \end{array} \\
 +
&  & \} & \\
 +
\end{array}</math>
 +
| style="border-left:1px solid black; text-align:center" |
 +
<math>\operatorname{F2.2c~:~R11c}</math></p>
 +
|- style="height:20px"
 +
| colspan="3" | &nbsp;
 +
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 +
|- style="height:100px"
 +
| &nbsp;
 +
| valign="top" | <math>\operatorname{F2.2d.}</math>
 +
| valign="top" |
 +
<math>\begin{array}{cccl}
 +
\upharpoonleft \operatorname{Der}^L \upharpoonright
 +
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
 +
&  &    & \begin{array}{cccl}
 +
          \underset{o \in O}{\operatorname{Conj}} \\
 +
          & \downharpoonleft  & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
 +
          &                  & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
 +
          &                  & ) & \\
 +
          & \downharpoonright &  & \\
 +
          \end{array} \\
 +
&  & \} & \\
 +
\end{array}</math>
 +
| style="border-left:1px solid black; text-align:center" |
 +
<math>\operatorname{F2.2d~:~Log}</math>
 +
|- style="height:20px"
 +
| colspan="3" | &nbsp;
 +
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 +
|- style="height:100px"
 +
| &nbsp;
 +
| valign="top" | <math>\operatorname{F2.2e.}</math>
 +
| valign="top" |
 +
<math>\begin{array}{cccl}
 +
\upharpoonleft \operatorname{Der}^L \upharpoonright
 +
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
 +
&  &    & \begin{array}{ccl}
 +
          \underset{o \in O}{\operatorname{Conj}} \\
 +
          & \underline{((} & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
 +
          & ,             & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
 +
          & \underline{))} & \\
 +
          \end{array} \\
 +
&  & \} & \\
 +
\end{array}</math>
 +
| style="border-left:1px solid black; text-align:center" |
 +
<math>\operatorname{F2.2e~:~Log}</math>
 +
|- style="height:20px"
 +
| colspan="3" | &nbsp;
 +
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 +
|- style="height:100px"
 +
| &nbsp;
 +
| valign="top" | <math>\operatorname{F2.2f.}</math>
 +
| valign="top" |
 +
<math>\begin{array}{cccl}
 +
\upharpoonleft \operatorname{Der}^L \upharpoonright
 +
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
 +
&  &    & \begin{array}{cll}
 +
          \underset{o \in O}{\operatorname{Conj}} \\
 +
          & \underline{((}    & \upharpoonleft \operatorname{Den}^L x \upharpoonright \\
 +
          & ,                 & \upharpoonleft \operatorname{Den}^L y \upharpoonright \\
 +
          & \underline{))}^\$ & (o) \\
 +
          \end{array} \\
 +
&  & \} & \\
 +
\end{array}</math>
 +
| style="border-left:1px solid black; text-align:center" |
 +
<math>\operatorname{F2.2f~:~$~}</math>
 +
|}
 +
|}
    
<br>
 
<br>
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/78810"

Navigation menu