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| | ====C<sub>3</sub>. Dominant form theorem==== | | ====C<sub>3</sub>. Dominant form theorem==== |
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| − | The third theorem to be proved here is one that GSB annotates as ''Integration'', but it may also be regarded as a matter of ''Dominance or Recession'' among forms. | + | The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence 3'' <math>(C_3)\!</math> or ''Integration''. A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs. |
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| − | o-----------------------------------------------------------o
| + | {| align="center" border="0" cellpadding="10" cellspacing="0" |
| − | | C_3.` Dominant Form Theorem ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| + | | [[Image:PERS_Figure_10.jpg|500px]] || (10) |
| − | o-----------------------------------------------------------o
| + | |} |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` a @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | o-----------------------------------------------------------o
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| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` a( )` ` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | o-----------------------------------------------------------o
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| − | | ` ` ` ` ` ` ` `Remark <---- | ----> Recess ` ` ` ` ` ` ` `|
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| − | o-----------------------------------------------------------o
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| − | Here is a proof of the ''Dominant Form Theorem''. | + | Here is a proof of the Dominant Form Theorem. |
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| − | o-----------------------------------------------------------o
| + | {| align="center" border="0" cellpadding="10" cellspacing="0" |
| − | | C_3.` Dominant Form Theorem.` Proof.` ` ` ` ` ` ` ` ` ` ` |
| + | | [[Image:PERS_Figure_11.jpg|500px]] || (11) |
| − | o-----------------------------------------------------------o
| + | |} |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | o=============================< C2. Regenerate "a" >========o
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| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | o=============================< J1. Delete "a" >============o
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| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| − | o=============================< QED >=======================o
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| − | If you scan the elementary steps that lead up to this point, you will notice two distinct qualities of the proofs so far:
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| − | * One brand of proof has that ''falling off a log and rolling downhill'' sort of quality that is earnestly to be wished for but seldom to be seen, at least, never so often as we'd wish.
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| − | * The other kind, more kith o' death than kind, has a quality strained past mercy, with a ''how in the heck did anybody ever think of that?'' sort of subtlety that all too unfortunately rules the roost whenever we begin to extend our practice to more and more compelling theories.
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| − | This is, to me, at least, a surprising observation, and though I have no grand conclusion to draw from it at the moment, it occurs to me that it might be a useful measure to keep in mind as we essay forth.
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| | ===Exemplary proofs=== | | ===Exemplary proofs=== |