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Directory:Jon Awbrey/Papers/Syntactic Transformations (view source)
Revision as of 15:15, 3 February 2009
, 15:15, 3 February 2009→Syntactic Transformation Rules: format rule box
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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%"
Rule 10
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
If X, Y c U,
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then the following are equivalent:
| width="20%" style="border-left:1px solid black" | <math>\operatorname{Rule~10}</math>
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R10a. X = Y. :D2a
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R10b. u C X <=> u C Y, for all u C U. :D2b
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R10c. [u C X] = [u C Y]. :R8b
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="60%" style="border-top:1px solid black" | <math>P, Q ~\subseteq~ X</math>
R10d. For all u C U,
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R10e. ConjUu ( [u C X](u) = [u C Y](u) ). :R8d
| <math>\text{then}\!</math>
| <math>\text{the following are equivalent:}\!</math>
R10f. ConjUu ( [u C X](u) <=> [u C Y](u) ). :R8e
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R10g. ConjUu (( [u C X](u) , [u C Y](u) )). :R8f
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R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g
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</pre>
|- style="height:40px"
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| width="18%" style="border-top:1px solid black" | <math>\operatorname{R10a.}</math>
| width="60%" style="border-top:1px solid black" | <math>P ~=~ Q</math>
| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{R10a~:~D2a}</math>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R10b.}</math>
| <math>\overset{X}{\underset{x}{\forall}}~ (x \in P ~\Leftrightarrow~ x \in Q)</math>
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<p><math>\operatorname{R10b~:~D2b}</math></p>
<p><math>\operatorname{R10b~:~R8a}</math></p>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R10c.}</math>
| <math>\downharpoonleft x \in P \downharpoonright ~=~ \downharpoonleft x \in Q \downharpoonright</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10c~:~R8b}</math>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R10d.}</math>
| <math>\overset{X}{\underset{x}{\forall}}~ \downharpoonleft x \in P \downharpoonright (x) ~=~ \downharpoonleft x \in Q \downharpoonright (x)</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10d~:~R8c}</math>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R10e.}</math>
| <math>\operatorname{Conj_x^X}~ (\downharpoonleft x \in P \downharpoonright (x) ~=~ \downharpoonleft x \in Q \downharpoonright (x))</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10e~:~R8d}</math>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R10f.}</math>
| <math>\operatorname{Conj_x^X}~ (\downharpoonleft x \in P \downharpoonright (x) ~\Leftrightarrow~ \downharpoonleft x \in Q \downharpoonright (x))</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10f~:~R8e}</math>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R10g.}</math>
| <math>\operatorname{Conj_x^X}~ \underline{((}~ \downharpoonleft x \in P \downharpoonright (x) ~,~ \downharpoonleft x \in Q \downharpoonright (x) ~\underline{))}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10g~:~R8f}</math>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R10h.}</math>
| <math>\operatorname{Conj_x^X}~ \underline{((}~ \downharpoonleft x \in P \downharpoonright ~,~ \downharpoonleft x \in Q \downharpoonright ~\underline{))}^\$ (x)</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10h~:~R8g}</math>
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