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Assume a well-founded relation $<$ over a set $S$ and a property $P$ on $S$ such that:

  1. $P$ holds for all minimal elements of $S$.
  2. For every $b \in P$ and $a < b$ we have: if $P(a)$ then there exists $c < b$ such that $P(c) \Rightarrow P(b)$.

Question: does it follow that $P(c)$ holds for all $c \in S$?

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    $\begingroup$ I rephrased the question. Is this what you are asking? (If not, you can undo the edit by going to an earlier revision.) $\endgroup$ – Andrej Bauer Feb 15 '17 at 15:04
  • $\begingroup$ I might be missing a subtle difference, but it seems to be what I am asking. $\endgroup$ – choeger Feb 15 '17 at 15:06
  • $\begingroup$ In that case, I have the answer. $\endgroup$ – Andrej Bauer Feb 15 '17 at 15:09
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As stated, the reasoning does not make much sense to me.

You claim that, in your specific case, you have

  • $a < b \wedge P(a) \Longrightarrow \exists c < b$
  • $P(a) \wedge P(b) \Longrightarrow P(b)$

but these properties are trivial tautologies -- they tell nothing about your specific case.

In the first one, if $a < b$, we already know $\exists c<b$. We don't even need $P(a)$.

The second one is an instance of the tautology $p \land q \implies q$.

I'm not sure how these two properties help in the induction proof at all.

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  • $\begingroup$ Thank you for pointing that out. The first part was an abuse of the existence quantor. What I meant was, that $c$ is actually an element of the set. The second was just a stupid typo. $\endgroup$ – choeger Feb 15 '17 at 14:14
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The answer is positive.

To show that $P$ holds for all $c \in S$, we need to establish, for all $b \in S$: $$(\forall a < b . P(a)) \Rightarrow P(b).$$ Then we may apply the well-foundedness of $<$ to $P$.

So consider any $b \in S$ and suppose $\forall a < b . P(a)$. We need to show that $P(b)$ holds. There are two cases:

  1. There is no $a < b$. In this case $b$ is minimal and so $P(b)$ by the first assumption in the statement of the question.

  2. There is some $a < b$. Then we know that $P(a)$ holds. By the second assumption of the statement of the question there is $c < b$ such that $P(c) \Rightarrow P(b)$. However, since $c < b$ we do have $P(c)$ by assumption, and therefore $P(b)$.

QED.

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