# mathematical induction

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## mathematical induction

n.
Induction.
American Heritage® Dictionary of the English Language, Fifth Edition. Copyright © 2016 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.
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Among their topics are elementary modifications for curves, interpolations and short exact sequences, a stronger inductive hypothesis, inductive arguments, and the three exceptional cases.
For the inductive hypothesis, let us assume that the result holds for k = n.
Note that 2m(m + 1) > (m + 1)(m + 2), from the inductive hypothesis we have [2.sup.m+1+1] > 2m(m + 1) > (m + 1)(m + 2).
Then the revolutionary inductive hypothesis of Charles Darwin late in the 19th century led to the reactionary, evolutionary vision that there was no evidence of beneficial design of any kind.
Then, by applying the inductive hypothesis, for all i, there exists an integer [n.sub.i] [is greater than or equal to] 0 such that: [[Eta].sub.i] [[is less than or equal to].sup.([n.sub.i]) [[Eta]'.sub.i].
From the inductive hypothesis, it follows that: [[[Eta].sub.i] [is less than or equal to] [[Eta]'.sub.i].
Hence, we can apply the inductive hypothesis and get: [[Eta]'.sub.i] = [[Eta].sub.i] for all i; that is, [[Tau].sub.1] = [[Tau].sub.2].
Hence, we can apply the inductive hypothesis and get that, for all i: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [r.sub.i] = max{[m.sub.i], [n.sub.i]} [is less than] r; that is, [[Tau].sub.1] [[is less than or equal to].sup.(r)] [[Tau].sub.3].
From the left premise ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), the right assumption ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), and the inductive hypothesis we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
From the premise ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), the constructed derivation ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), and the inductive hypothesis we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
From the premise ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), the right assumption ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), and the inductive hypothesis we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Since the trace between these states is balanced we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by the inductive hypothesis. The state after [app.sub.2] is ([cc.sub.[Lambda]], [Delta], e'[x/y], S, [Theta]'), and the trace of ([cc.sub.[Lambda]], [Delta], e'[x/y], S, [Theta]') [[right arrow].sup.*] ([cc.sub.1], [[Gamma].sub.1], z, S, [[Theta].sub.1]) is balanced; so we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the inductive hypothesis.

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