Fraenkel word: Difference between revisions

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In case 1, by the preceding lemma {{!}}''u''{{!}}'''i''' = {{!}}''u''{{!}}/2''i''+1 and {{!}}''v''{{!}}'''i''' = {{!}}''v''{{!}}/2''i''+1, and hence {{!}}''w''{{!}}'''i''' = {{!}}''w''{{!}}/2''i''+1 = ceil({{!}}''w''{{!}}/2''i''+1).

In case 2, suppose ''w'' = ''ustv'' where ''st'' is as in case 1 and {{!}}''u''{{!}} and {{!}}''v''{{!}} are less than 2''i''+1. Neither ''u'' nor ''v'' can contain an '''i''', as they are subwords of ''F''''i''. ~~Since~~

In case 2, suppose ''w'' = ''ustv'' where ''st'' is as in case 1 and {{!}}''u''{{!}} and {{!}}''v''{{!}} are less than 2''i''+1. Neither ''u'' nor ''v'' can contain an '''i''', as they are subwords of ''F''''i''. Using {{!}}''st''{{!}} = {{!}}''st''{{!}}'''i'''2''i+1, we have

{{!}}''st''{{!}} < {{!}}''w''{{!}} = {{!}}''st''{{!}} + {{!}}''u''{{!}} + {{!}}''v''{{!}} ≤ {{!}}''st''{{!}} + 2''i''+1 − 2,

{{!}}''st''{{!}}'''i'''2''i+1 < {{!}}''w''{{!}} = {{!}}''st''{{!}}'''i'''2''i+1 + {{!}}''u''{{!}} + {{!}}''v''{{!}} ≤ {{!}}''st''{{!}}'''i'''2''i''+1+ 2''i''+1 − 2,

~~we have~~

thus

{{!}}''w''{{!}}'''i''' ≥ {{!}}''st''{{!}}'''i''' = {{!}}''st''{{!}}/2''i''+1 = ceil({{!}}''w''{{!}}/2''i''+1) − 1.

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 In case 1, by the preceding lemma {{!}}''u''{{!}}<sub>'''i'''</sub> = {{!}}''u''{{!}}/2<sup>''i''+1</sup> and {{!}}''v''{{!}}<sub>'''i'''</sub> = {{!}}''v''{{!}}/2<sup>''i''+1</sup>, and hence {{!}}''w''{{!}}<sub>'''i'''</sub> = {{!}}''w''{{!}}/2<sup>''i''+1</sup> = ceil({{!}}''w''{{!}}/2<sup>''i''+1</sup>).
-In case 2, suppose ''w'' = ''ustv'' where ''st'' is as in case 1 and {{!}}''u''{{!}} and {{!}}''v''{{!}} are less than 2<sup>''i''+1</sup>. Neither ''u'' nor ''v'' can contain an '''i''', as they are subwords of ''F''<sub>''i''</sub>. Since
+In case 2, suppose ''w'' = ''ustv'' where ''st'' is as in case 1 and {{!}}''u''{{!}} and {{!}}''v''{{!}} are less than 2<sup>''i''+1</sup>. Neither ''u'' nor ''v'' can contain an '''i''', as they are subwords of ''F''<sub>''i''</sub>. Using {{!}}''st''{{!}} = {{!}}''st''{{!}}<sub>'''i'''</sub>2<sup>''i+1</sup>, we have
-{{!}}''st''{{!}} < {{!}}''w''{{!}} = {{!}}''st''{{!}} + {{!}}''u''{{!}} + {{!}}''v''{{!}} &le; {{!}}''st''{{!}} +  2<sup>''i''+1</sup> &minus; 2,
+{{!}}''st''{{!}}<sub>'''i'''</sub>2<sup>''i+1</sup> < {{!}}''w''{{!}} = {{!}}''st''{{!}}<sub>'''i'''</sub>2<sup>''i+1</sup> + {{!}}''u''{{!}} + {{!}}''v''{{!}} &le; {{!}}''st''{{!}}<sub>'''i'''</sub>2<sup>''i''+1</sup>+  2<sup>''i''+1</sup> &minus; 2,
-we have
+thus
 {{!}}''w''{{!}}<sub>'''i'''</sub> &ge; {{!}}''st''{{!}}<sub>'''i'''</sub> = {{!}}''st''{{!}}/2<sup>''i''+1</sup> = ceil({{!}}''w''{{!}}/2<sup>''i''+1</sup>) &minus; 1.