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

Revision as of 15:00, 29 January 2009

359 bytes added ,  15:00, 29 January 2009
→‎Syntactic Transformations: segment
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A condition is ''amenable'' to a rule if any of its conceivable expressions formally match any of the expressions that are enumerated by the rule.  Further, it requires the relegation of the other expressions to the production of a result.  Thus, there is the choice of an initial expression that needs to be checked on input for whether it fits the antecedent condition and there are several types of output that are generated as a consequence, only a few of which are usually needed at any given time.
 
A condition is ''amenable'' to a rule if any of its conceivable expressions formally match any of the expressions that are enumerated by the rule.  Further, it requires the relegation of the other expressions to the production of a result.  Thus, there is the choice of an initial expression that needs to be checked on input for whether it fits the antecedent condition and there are several types of output that are generated as a consequence, only a few of which are usually needed at any given time.
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'''Editing Note.'''  Need a transition here.  Give a brief description of the Tables of Translation rules that have now been moved to the Appendices, and then move on to the rest of the Definitions and Proof Schemata.
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'''Editing Note.'''  Need a transition here.  Give a brief description of the Tables of Translation Rules that have now been moved to the Appendices, and then move on to the rest of the Definitions and Proof Schemata.
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<pre>
 
<pre>
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V1c. (( v , w )).
 
V1c. (( v , w )).
 
</pre>
 
</pre>
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<pre>
 
<pre>
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V1c. (( v , w ))
 
V1c. (( v , w ))
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</pre>
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<br>
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<pre>
 
Value Rule 1
 
Value Rule 1
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V1c. (( f(u) , g(u) ))
 
V1c. (( f(u) , g(u) ))
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</pre>
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Value Rule 1
 
Value Rule 1
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V1c. (( f , g ))$
 
V1c. (( f , g ))$
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</pre>
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<pre>
 
Evaluation Rule 1
 
Evaluation Rule 1
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::
 
::
 
E1d. (( f , g ))$(u). :$1b
 
E1d. (( f , g ))$(u). :$1b
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</pre>
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<pre>
 
Evaluation Rule 1
 
Evaluation Rule 1
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::
 
::
 
E1d. (( f , g ))$(u). :$1b
 
E1d. (( f , g ))$(u). :$1b
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</pre>
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<pre>
 
Definition 2
 
Definition 2
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D2b. u C X  <=>  u C Y, for all u C U.
 
D2b. u C X  <=>  u C Y, for all u C U.
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<pre>
 
Definition 3
 
Definition 3
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D3b. f(u) = g(u), for all u C U.
 
D3b. f(u) = g(u), for all u C U.
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</pre>
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<pre>
 
Definition 4
 
Definition 4
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D4b. {<u, v> C UxB : v = [u C X]}
 
D4b. {<u, v> C UxB : v = [u C X]}
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</pre>
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<pre>
 
Definition 5
 
Definition 5
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: f(u) = [u C X], for all u C U.
 
: f(u) = [u C X], for all u C U.
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</pre>
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Given an indexed set of sentences, Sj for j C J, it is possible to consider the logical conjunction of the corresponding propositions.  Various notations for this concept are be useful in various contexts, a sufficient sample of which are recorded in Definition 6.
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Given an indexed set of sentences, <math>s_j\!</math> for <math>j \in J,</math> it is possible to consider the logical conjunction of the corresponding propositions.  Various notations for this concept are be useful in various contexts, a sufficient sample of which are recorded in Definition&nbsp;6.
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<pre>
 
Definition 6
 
Definition 6
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D6e. ConjJj Sj.
 
D6e. ConjJj Sj.
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</pre>
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<pre>
 
Definition 7
 
Definition 7
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R5e. {X} = {Y}. :D5a
 
R5e. {X} = {Y}. :D5a
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<pre>
 
Rule 6
 
Rule 6
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::
 
R6c. ConjUu (f(u) = g(u)). :D6e
 
R6c. ConjUu (f(u) = g(u)). :D6e
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<pre>
 
Rule 7
 
Rule 7
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::
 
R7f. ConjUu (( P , Q ))$(u). :$1b
 
R7f. ConjUu (( P , Q ))$(u). :$1b
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</pre>
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<pre>
 
Rule 8
 
Rule 8
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For instance, the observation that expresses the equality of sets in terms of their indicator functions can be formalized according to the pattern in Rule 9, namely, at lines (a, b, c), and these components of Rule 9 can be cited in future uses as "R9a", "R9b", "R9c", respectively.  Using Rule 7, annotated as "R7", to adduce a few properties of indicator functions to the account, it is possible to extend Rule 9 by another few steps, referenced as "R9d", "R9e", "R9f", "R9g".
 
For instance, the observation that expresses the equality of sets in terms of their indicator functions can be formalized according to the pattern in Rule 9, namely, at lines (a, b, c), and these components of Rule 9 can be cited in future uses as "R9a", "R9b", "R9c", respectively.  Using Rule 7, annotated as "R7", to adduce a few properties of indicator functions to the account, it is possible to extend Rule 9 by another few steps, referenced as "R9d", "R9e", "R9f", "R9g".
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<pre>
 
Rule 9
 
Rule 9
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::
 
R9g. ConjUu (( {X} , {Y} ))$(u). :R7f
 
R9g. ConjUu (( {X} , {Y} ))$(u). :R7f
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<br>
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<pre>
 
Rule 10
 
Rule 10
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::
 
R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g
 
R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g
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</pre>
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<br>
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<pre>
 
Rule 11
 
Rule 11
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Applying Rule 9, Rule 8, and the Logical Rules to the special case where S <=> (X = Y), one obtains the following general fact.
 
Applying Rule 9, Rule 8, and the Logical Rules to the special case where S <=> (X = Y), one obtains the following general fact.
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</pre>
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<br>
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<pre>
 
Fact 1
 
Fact 1
  
Jon Awbrey
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