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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
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<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 2.2
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|
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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If R c OxSxI,
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|- style="height:50px; text-align:center"
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| style="width:80%" | &nbsp;
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then the following are equivalent:
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| style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~2.2}</math>
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|}
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F2.2a. DerR = {<x, y> C SxI :
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|-
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Conj(o C O)
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|
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{Den(R, x)}(o) =
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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{Den(R, y)}(o)
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|- style="height:50px"
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} :R11a
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| width="2%"  style="border-top:1px solid black" | &nbsp;
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::
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| width="12%" style="border-top:1px solid black" | <math>\text{If}\!</math>
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F2.2b. {DerR} = { {<x, y> C SxI :
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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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Conj(o C O)
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| width="20%" style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
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{Den(R, x)}(o) =
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|- style="height:50px"
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{Den(R, y)}(o)
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| &nbsp;
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}
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| <math>\text{then}\!</math>
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} :R11b
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| <math>\text{the following are equivalent:}\!</math>
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::
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| style="border-left:1px solid black" | &nbsp;
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F2.2c. {DerR} c SxIxB
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|}
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:
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|-
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{DerR} = {<x, y, v> C SxIxB :
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|
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v =
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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[ Conj(o C O)
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|- style="height:10px"
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{Den(R, x)}(o) =
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| width="2%"  style="border-top:1px solid black" | &nbsp;
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{Den(R, y)}(o)
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| width="12%" style="border-top:1px solid black" | &nbsp;
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]
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| width="66%" style="border-top:1px solid black" | &nbsp;
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} :R11c
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| width="20%" style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
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::
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|- style="height:100px"
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F2.2d. {DerR} = {<x, y, v> C SxIxB :
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| &nbsp;
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v =
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| valign="top" | <math>\operatorname{F2.2a.}</math>
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Conj(o C O)
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| valign="top" |
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[ {Den(R, x)}(o) =
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<math>\begin{array}{cccl}
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{Den(R, y)}(o)
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\operatorname{Der}^L
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]
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& = & \{ & (x, y) \in S \times I ~: \\
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} :Log
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&  &    & \begin{array}{ccl}
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          \underset{o \in O}{\operatorname{Conj}} \\
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F2.2e. {DerR} = {<x, y, v> C SxIxB :
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          & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
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v =
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          & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
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Conj(o C O)
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          & ) & \\
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(( {Den(R, x)}(o),
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          \end{array} \\
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{Den(R, y)}(o)
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&  & \} & \\
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))
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\end{array}</math>
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} :Log
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.2a~:~R11a}</math>
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|- style="height:20px"
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F2.2f. {DerR} = {<x, y, v> C SxIxB :
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| colspan="3" | &nbsp;
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v =
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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Conj(o C O)
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|- style="height:100px"
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(( {Den(R, x)},
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| &nbsp;
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{Den(R, y)}
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| valign="top" | <math>\operatorname{F2.2b.}</math>
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))$(o)
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| valign="top" |
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} :$
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<math>\begin{array}{ccccl}
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</pre>
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\upharpoonleft \operatorname{Der}^L \upharpoonright
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& = & \upharpoonleft  & \{ & (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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&  & \upharpoonright &    & \\
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\end{array}</math>
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| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F2.2b~:~R11b}</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{F2.2c.}</math>
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| valign="top" |
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<math>\begin{array}{cccl}
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\upharpoonleft \operatorname{Der}^L \upharpoonright
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& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
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&  &    & \begin{array}{cccl}
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          \downharpoonleft  & \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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          \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" |
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<math>\operatorname{F2.2c~:~R11c}</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{F2.2d.}</math>
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| valign="top" |
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<math>\begin{array}{cccl}
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\upharpoonleft \operatorname{Der}^L \upharpoonright
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& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
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&  &    & \begin{array}{cccl}
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          \underset{o \in O}{\operatorname{Conj}} \\
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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" |
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<math>\operatorname{F2.2d~:~Log}</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{F2.2e.}</math>
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| valign="top" |
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<math>\begin{array}{cccl}
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\upharpoonleft \operatorname{Der}^L \upharpoonright
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& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
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&  &    & \begin{array}{ccl}
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          \underset{o \in O}{\operatorname{Conj}} \\
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          & \underline{((} & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
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          & ,             & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
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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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<math>\operatorname{F2.2e~:~Log}</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{F2.2f.}</math>
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| valign="top" |
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<math>\begin{array}{cccl}
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\upharpoonleft \operatorname{Der}^L \upharpoonright
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& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
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&  &    & \begin{array}{cll}
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          \underset{o \in O}{\operatorname{Conj}} \\
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          & \underline{((}    & \upharpoonleft \operatorname{Den}^L x \upharpoonright \\
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          & ,                 & \upharpoonleft \operatorname{Den}^L y \upharpoonright \\
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          & \underline{))}^\$ & (o) \\
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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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<math>\operatorname{F2.2f~:~$~}</math>
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|}
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|}
    
<br>
 
<br>
Jon Awbrey
12,122

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