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

edits

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