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

Revision as of 01:25, 4 February 2009

4,431 bytes added ,  01:25, 4 February 2009
→‎Syntactic Transformation Rules: 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 1
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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~1}</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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| style="border-top:1px solid black; width:2%"  | &nbsp;
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| style="border-top:1px solid black; width:18%" | <math>\text{If}\!</math>
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| style="border-top:1px solid black; width:60%" | <math>P, Q ~\subseteq~ X</math>
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| 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:60px"
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| style="width:2%;  border-top:1px solid black" | &nbsp;
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| style="width:14%; border-top:1px solid black" | <math>\operatorname{F1a.}</math>
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| style="width:64%; border-top:1px solid black" | <math>s \quad \Leftrightarrow \quad P ~=~ Q</math>
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| style="width:20%; border-top:1px solid black; border-left:1px solid black; text-align:center" |
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<math>\operatorname{F1a~:~R9a}</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:40px"
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| &nbsp;
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| <math>\operatorname{F1b.}</math>
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| <math>s \quad \Leftrightarrow \quad \upharpoonleft P \upharpoonright ~=~ \upharpoonleft Q \upharpoonright</math>
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| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F1b~:~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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| <math>\operatorname{F1c.}</math>
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| <math>s \quad \Leftrightarrow \quad \overset{X}{\underset{x}{\forall}}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math>
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| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F1c~:~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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| <math>\operatorname{F1d.}</math>
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| <math>s \quad \Leftrightarrow \quad \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math>
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| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{F1d~:~R9d}</math></p>
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<p><math>\operatorname{F1d~:~R8a}</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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| <math>\operatorname{F1e.}</math>
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| <math>\downharpoonleft s \downharpoonright \quad = \quad \downharpoonleft \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright</math>
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| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{F1e~:~R8b}</math></p>
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<p><math>\operatorname{F1e~:~\_\_?\_\_}</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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| <math>\operatorname{F1f.}</math>
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| <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \downharpoonleft (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright</math>
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| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{F1f~:~\_\_?\_\_}</math></p>
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<p><math>\operatorname{F1f~:~\_\_?\_\_}</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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| <math>\operatorname{F1g.}</math>
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| <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright (x) ~,~ \upharpoonleft Q \upharpoonright (x) ~\underline{))}</math>
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| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{F1g~:~\_\_?\_\_}</math></p>
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<p><math>\operatorname{F1g~:~\_\_?\_\_}</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:40px"
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| &nbsp;
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| <math>\operatorname{F1h.}</math>
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| <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright ~\underline{))}^\$ (x)</math>
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| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F1g~:~~\_\_?\_\_}</math>
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|}
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|}
   −
If X,Y c U,
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<br>
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then the following are equivalent:
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F1a. S <=> X = Y. :R9a
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::
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F1b. S <=> {X} = {Y}. :R9b
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::
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F1c. S <=> {X}(u) = {Y}(u), for all u C U. :R9c
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::
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F1d. S <=> ConjUu ( {X}(u) = {Y}(u) ). :R9d
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:R8a
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::
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F1e. [S] = [ ConjUu ( {X}(u) = {Y}(u) ) ]. :R8b
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:???
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::
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F1f. [S] = ConjUu [ {X}(u) = {Y}(u) ]. :???
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::
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F1g. [S] = ConjUu (( {X}(u) , {Y}(u) )). :$1a
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::
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F1h. [S] = ConjUu (( {X} , {Y} ))$(u). :$1b
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///
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{u C U : (f, g)$(u)}
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= {u C U : (f(u), g(u))}
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= {u C
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///
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</pre>
      
===Derived Equivalence Relations===
 
===Derived Equivalence Relations===
Jon Awbrey
12,080

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