Consistency: Difference between revisions
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If for any interval ''r'', ''T'' (''r'') is the closest approximation to ''r'' in ''T'', and if ''V'' (''r'') is ''r'' mapped by a val ''V'', then ''T'' is consistent with respect to a set of intervals ''S'' if there exists a val ''V'' such that ''T'' (''r'') = ''V'' (''r'') for any ''r'' in ''S''. | If for any interval ''r'', ''T'' (''r'') is the closest approximation to ''r'' in ''T'', and if ''V'' (''r'') is ''r'' mapped by a val ''V'', then ''T'' is consistent with respect to a set of intervals ''S'' if there exists a val ''V'' such that ''T'' (''r'') = ''V'' (''r'') for any ''r'' in ''S''. | ||
{{ | {{Proof | ||
contents=Let us denote the monzo of any ratio ''r'' by '''m'''. Due to the linearity of the interval space, for any intervals ''r''<sub>''i''</sub>, ''r''<sub>''j''</sub>, and ''r''<sub>''i''</sub>''r''<sub>''j''</sub> in ''S'', their monzos are '''m'''<sub>''i''</sub>, '''m'''<sub>''j''</sub>, and '''m'''<sub>''i''</sub> + '''m'''<sub>''j''</sub>, respectively. | | title=Proof for equivalence | ||
| contents=Let us denote the monzo of any ratio ''r'' by '''m'''. Due to the linearity of the interval space, for any intervals ''r''<sub>''i''</sub>, ''r''<sub>''j''</sub>, and ''r''<sub>''i''</sub>''r''<sub>''j''</sub> in ''S'', their monzos are '''m'''<sub>''i''</sub>, '''m'''<sub>''j''</sub>, and '''m'''<sub>''i''</sub> + '''m'''<sub>''j''</sub>, respectively. | |||
The ratio ''r'' mapped by the val ''V'' is the tempered step number ''V'' (''r'') = ''V''·'''m''', with the following identity: | The ratio ''r'' mapped by the val ''V'' is the tempered step number ''V'' (''r'') = ''V''·'''m''', with the following identity: | ||