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

Revision as of 04:18, 13 February 2009

5,200 bytes added ,  04:18, 13 February 2009
→‎Derived Equivalence Relations: format fact box
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<br>
 
<br>
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<pre>
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{| 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%"
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Fact 2.3
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|
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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|- style="height:50px; text-align:center"
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| style="width:80%" | &nbsp;
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| style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~2.3}</math>
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|}
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|-
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|
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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|- style="height:50px"
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| width="2%"  style="border-top:1px solid black" | &nbsp;
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| width="12%" style="border-top:1px solid black" | <math>\text{If}\!</math>
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| width="66%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math>
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| width="20%" style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
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|- style="height:50px"
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| &nbsp;
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| <math>\text{then}\!</math>
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| <math>\text{the following are equivalent:}\!</math>
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| style="border-left:1px solid black" | &nbsp;
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|}
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|-
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|
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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|- style="height:10px"
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| width="2%"  style="border-top:1px solid black" | &nbsp;
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| width="12%" style="border-top:1px solid black" | &nbsp;
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| width="66%" style="border-top:1px solid black" | &nbsp;
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| width="20%" style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
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|- style="height:100px"
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| &nbsp;
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| valign="top" | <math>\operatorname{F2.3a.}</math>
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| valign="top" |
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<math>\begin{array}{cccl}
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\operatorname{Der}^L
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& = & \{ & (x, y) \in S \times I ~: \\
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&  &    & \begin{array}{ccl}
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          \underset{o \in O}{\operatorname{Conj}} \\
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          & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
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          & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
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          & ) & \\
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          \end{array} \\
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&  & \} & \\
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\end{array}</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.3a~:~R11a}</math>
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|- style="height:20px"
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| colspan="3" | &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:100px"
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| &nbsp;
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| valign="top" | <math>\operatorname{F2.3b.}</math>
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| valign="top" |
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<math>\begin{array}{ccccl}
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\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
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& = & \downharpoonleft  & \underset{o \in O}{\operatorname{Conj}} \\
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&  &                  & & \begin{array}{cl}
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                            ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
 +
                            = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
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                            ) & \\
 +
                            \end{array} \\
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&  & \downharpoonright & & \\
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\end{array}</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.3b~:~R11d}</math>
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|- style="height:20px"
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| colspan="3" | &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:100px"
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| &nbsp;
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| valign="top" | <math>\operatorname{F2.3c.}</math>
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| valign="top" |
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<math>\begin{array}{ccccl}
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\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
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& = & \underset{o \in O}{\operatorname{Conj}} \\
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&  & & \begin{array}{ccl}
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        \downharpoonleft  & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
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                          & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
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                          & ) & \\
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        \downharpoonright &  & \\
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        \end{array} \\
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&  & & \\
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\end{array}</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.3c~:~Log}</math></p>
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|- style="height:20px"
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| colspan="3" | &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:100px"
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| &nbsp;
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| valign="top" | <math>\operatorname{F2.3d.}</math>
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| valign="top" |
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<math>\begin{array}{ccccl}
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\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
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& = & \underset{o \in O}{\operatorname{Conj}} \\
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&  & & \begin{array}{ccl}
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        \downharpoonleft  & ( & \upharpoonleft \operatorname{Den}^L \upharpoonright (o, x) \\
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                          & = & \upharpoonleft \operatorname{Den}^L \upharpoonright (o, y) \\
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                          & ) & \\
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        \downharpoonright &  & \\
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        \end{array} \\
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&  & & \\
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\end{array}</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.3d~:~Def}</math>
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|- style="height:20px"
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| colspan="3" | &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:100px"
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| &nbsp;
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| valign="top" | <math>\operatorname{F2.3e.}</math>
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| valign="top" |
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<math>\begin{array}{ccccl}
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\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
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& = & \underset{o \in O}{\operatorname{Conj}} \\
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&  & & \begin{array}{cl}
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        \underline{((} & \upharpoonleft \operatorname{Den}^L \upharpoonright (o, x) \\
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        ,              & \upharpoonleft \operatorname{Den}^L \upharpoonright (o, y) \\
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        \underline{))} & \\
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        \end{array} \\
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&  & & \\
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\end{array}</math>
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| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{F2.3e~:~Log}</math></p>
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<p><math>\operatorname{F2.3e~:~D10b}</math></p>
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|- style="height:20px"
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| colspan="3" | &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:100px"
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| &nbsp;
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| valign="top" | <math>\operatorname{F2.3f.}</math>
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| valign="top" |
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<math>\begin{array}{ccccl}
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\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
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& = & \underset{o \in O}{\operatorname{Conj}} \\
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&  & & \begin{array}{cl}
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        \underline{((} & \upharpoonleft L_{OS} \upharpoonright (o, x) \\
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        ,              & \upharpoonleft L_{OS} \upharpoonright (o, y) \\
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        \underline{))} & \\
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        \end{array} \\
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&  & & \\
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\end{array}</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.3f~:~D10a}</math>
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|}
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|}
   −
If R c OxSxI,
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<br>
−
 
  −
then the following are equivalent:
  −
 
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F2.3a. DerR = {<x, y> C SxI :
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Conj(o C O)
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{Den(R, x)}(o) =
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{Den(R, y)}(o)
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} :R11a
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::
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F2.3b. {DerR} : SxI �> B
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:
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{DerR}(x, y) = [ Conj(o C O)
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{Den(R, x)}(o) =
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{Den(R, y)}(o)
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] :R11d
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::
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F2.3c. {DerR}(x, y) = Conj(o C O)
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[ {Den(R, x)}(o) =
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{Den(R, y)}(o)
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] :Log
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::
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F2.3d. {DerR}(x, y) = Conj(o C O)
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[ {DenR}(o, x) =
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{DenR}(o, y)
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] :Def
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::
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F2.3e. {DerR}(x, y) = Conj(o C O)
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(( {DenR}(o, x),
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{DenR}(o, y)
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)) :Log
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:D10b
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::
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F2.3f. {DerR}(x, y) = Conj(o C O)
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(( {ROS}(o, x),
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{ROS}(o, y)
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)) :D10a
  −
</pre>
      
===Digression on Derived Relations===
 
===Digression on Derived Relations===
Jon Awbrey
12,080

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