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Directory:Jon Awbrey/Papers/Syntactic Transformations (view source)
Revision as of 14:36, 3 February 2009
, 14:36, 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 9
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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~9}</math>
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R9a. X = Y. :R5a
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R9b. {X} = {Y}. :R5e
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R9c. {X}(u) = {Y}(u), for all u C U. :R7b
| 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>
R9d. ConjUu ( {X}(u) = {Y}(u) ). :R7c
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R9e. ConjUu ( {X}(u) <=> {Y}(u) ). :R7d
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| <math>\text{then}\!</math>
R9f. ConjUu (( {X}(u) , {Y}(u) )). :R7e
| <math>\text{the following are equivalent:}\!</math>
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R9g. ConjUu (( {X} , {Y} ))$(u). :R7f
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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| width="18%" style="border-top:1px solid black" | <math>\operatorname{R9a.}</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{R9a~:~R5a}</math>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R9b.}</math>
| <math>\upharpoonleft P \upharpoonright ~=~ \upharpoonleft Q \upharpoonright</math>
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<p><math>\operatorname{R9b~:~R5e}</math></p>
<p><math>\operatorname{R9b~:~R7a}</math></p>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R9c.}</math>
| <math>\overset{X}{\underset{x}{\forall}}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R9c~:~R7b}</math>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R9d.}</math>
| <math>\operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R9d~:~R7c}</math>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R9e.}</math>
| <math>\operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~\Leftrightarrow~ \upharpoonleft Q \upharpoonright (x))</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R9e~:~R7d}</math>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R9f.}</math>
| <math>\operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright (x) ~,~ \upharpoonleft Q \upharpoonright (x) ~\underline{))}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R9f~:~R7e}</math>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R9g.}</math>
| <math>\operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright ~\underline{))}^\$ (x)</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R9g~:~R7f}</math>
|}
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