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Directory:Jon Awbrey/Papers/Cactus Language (view source)

Revision as of 02:32, 29 January 2009

5,240 bytes added ,  02:32, 29 January 2009
→‎Syntactic Transformations: format table
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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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Geometric Translation Rule 2
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|
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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If X, Y c U
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|- style="height:48px; text-align:right"
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| width="98%" | <math>\text{Geometric Translation Rule 2}\!</math>
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and P, Q U -> B, such that:
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| width="2%"  | &nbsp;
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|}
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G2a. {X} = P  and {Y} = Q,
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|-
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|
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then the following equations hold:
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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|- style="height:48px"
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G2b00. {{}} = () = 0 : U->B.
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| width="2%"  style="border-top:1px solid black" | &nbsp;
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| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
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G2b01. {~X n ~Y} = ({X})({Y}) = (P)(Q).
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| width="84%" style="border-top:1px solid black" | <math>P, Q \subseteq X</math>
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|- style="height:48px"
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G2b02. {~X n Y} = ({X}){Y} = (P) Q.
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| &nbsp;
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| <math>\text{and}\!</math>
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G2b03. {~X} = ({X}) = (P).
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| <math>p, q ~:~ X \to \underline\mathbb{B}</math>
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|- style="height:48px"
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G2b04. {X n ~Y} = {X}({Y}) = P (Q).
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| &nbsp;
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| <math>\text{such that:}\!</math>
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G2b05. {~Y} = ({Y}) = (Q).
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| &nbsp;
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|- style="height:48px"
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G2b06. {X + Y} = ({X}, {Y}) = (P, Q).
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| &nbsp;
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| <math>\text{G2a.}\!</math>
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G2b07. {~(X n Y)} = ({X}.{Y}) = (P Q).
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| <math>\upharpoonleft P \upharpoonright ~=~ p \quad \operatorname{and} \quad \upharpoonleft Q \upharpoonright ~=~ q</math>
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|- style="height:48px"
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G2b08. {X n Y} = {X}.{Y} = P.Q.
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| &nbsp;
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| <math>\text{then}\!</math>
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G2b09. {~(X + Y)} = (({X}, {Y})) = ((P, Q)).
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| <math>\text{the following equations hold:}\!</math>
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|}
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G2b10. {Y} = {Y} = Q.
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|-
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|
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G2b11. {~(X n ~Y)} = ({X}({Y})) = (P (Q)).
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{| align="center" cellpadding="0" cellspacing="0" style="text-align:center" width="100%"
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|- style="height:52px"
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G2b12. {X} = {X} = P.
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| width="2%"  style="border-top:1px solid black" | &nbsp;
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| width="14%" style="border-top:1px solid black" align="left" | <math>\text{G2b}_{0}.\!</math>
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G2b13. {~(~X n Y)} = (({X}) {Y}) = ((P) Q).
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| width="32%" style="border-top:1px solid black" |
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<math>\upharpoonleft \varnothing \upharpoonright</math>
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G2b14. {X u Y} = (({X})({Y})) = ((P)(Q)).
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| width="4%"  style="border-top:1px solid black" | <math>=\!</math>
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| width="28%" style="border-top:1px solid black" | <math>(~)</math>
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G2b15. {U} = (()) = 1 : U->B.
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| width="4%"  style="border-top:1px solid black" | <math>=\!</math>
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</pre>
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| width="16%" style="border-top:1px solid black" | <math>(~)</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{G2b}_{1}.\!</math>
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| <math>\upharpoonleft \overline{P} ~\cap~ \overline{Q} \upharpoonright</math>
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| <math>=\!</math>
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| <math>(\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright)</math>
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| <math>=\!</math>
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| <math>(p)(q)\!</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{G2b}_{2}.\!</math>
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| <math>\upharpoonleft \overline{P} ~\cap~ Q \upharpoonright</math>
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| <math>=\!</math>
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| <math>(\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright</math>
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| <math>=\!</math>
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| <math>(p) q\!</math>
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|- style="height:52px"
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| &nbsp;
 +
| align="left" | <math>\text{G2b}_{3}.\!</math>
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| <math>\upharpoonleft \overline{P} \upharpoonright</math>
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| <math>=\!</math>
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| <math>(\upharpoonleft P \upharpoonright)</math>
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| <math>=\!</math>
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| <math>(p)\!</math>
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|- style="height:52px"
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| &nbsp;
 +
| align="left" | <math>\text{G2b}_{4}.\!</math>
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| <math>\upharpoonleft P ~\cap~ \overline{Q} \upharpoonright</math>
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| <math>=\!</math>
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| <math>\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright)</math>
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| <math>=\!</math>
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| <math>p (q)\!</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{G2b}_{5}.\!</math>
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| <math>\upharpoonleft \overline{Q} \upharpoonright</math>
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| <math>=\!</math>
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| <math>(\upharpoonleft Q \upharpoonright)</math>
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| <math>=\!</math>
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| <math>(q)\!</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{G2b}_{6}.\!</math>
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| <math>\upharpoonleft P ~+~ Q \upharpoonright</math>
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| <math>=\!</math>
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| <math>(\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright)</math>
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| <math>=\!</math>
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| <math>(p, q)\!</math>
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|- style="height:52px"
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| &nbsp;
 +
| align="left" | <math>\text{G2b}_{7}.\!</math>
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| <math>\upharpoonleft \overline{P ~\cap~ Q} \upharpoonright</math>
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| <math>=\!</math>
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| <math>(\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright)</math>
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| <math>=\!</math>
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| <math>(p q)\!</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{G2b}_{8}.\!</math>
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| <math>\upharpoonleft P ~\cap~ Q \upharpoonright</math>
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| <math>=\!</math>
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| <math>\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright</math>
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| <math>=\!</math>
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| <math>p q\!</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{G2b}_{9}.\!</math>
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| <math>\upharpoonleft \overline{P ~+~ Q} \upharpoonright</math>
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| <math>=\!</math>
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| <math>((\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright))</math>
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| <math>=\!</math>
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| <math>((p, q))\!</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{G2b}_{10}.\!</math>
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| <math>\upharpoonleft Q \upharpoonright</math>
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| <math>=\!</math>
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| <math>\upharpoonleft Q \upharpoonright</math>
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| <math>=\!</math>
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| <math>q\!</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{G2b}_{11}.\!</math>
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| <math>\upharpoonleft \overline{P ~\cap~ \overline{Q}} \upharpoonright</math>
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| <math>=\!</math>
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| <math>(\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright))</math>
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| <math>=\!</math>
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| <math>(p (q))\!</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{G2b}_{12}.\!</math>
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| <math>\upharpoonleft P \upharpoonright</math>
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| <math>=\!</math>
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| <math>\upharpoonleft P \upharpoonright</math>
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| <math>=\!</math>
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| <math>p\!</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{G2b}_{13}.\!</math>
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| <math>\upharpoonleft \overline{\overline{P} ~\cap~ Q} \upharpoonright</math>
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| <math>=\!</math>
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| <math>((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright)</math>
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| <math>=\!</math>
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| <math>((p) q)\!</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{G2b}_{14}.\!</math>
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| <math>\upharpoonleft P ~\cup~ Q \upharpoonright</math>
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| <math>=\!</math>
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| <math>((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright))</math>
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| <math>=\!</math>
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| <math>((p)(q))\!</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{G2b}_{15}.\!</math>
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| <math>\upharpoonleft X \upharpoonright</math>
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| <math>=\!</math>
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| <math>((~))</math>
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| <math>=\!</math>
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| <math>((~))</math>
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|}
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|}
    
<br>
 
<br>
Jon Awbrey
12,080

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