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Aug 16, 2022 at 18:45 comment added Wouter Zandsteeg I also came up with a way to avoid what I think is using the theorem itself, but it doesn't involve the latter four inequalities at all. For the first one I did: $x^R+y\leq x+z$, so none of $x^R+y\geq (x+z)^R$, and thus $x^R+y \ngeq x^R+z$, but from $y\geq z$ and induction follows $x^R+y \geq x^R+z$, so a contradiction. The third one is similar. The second one implies by $y^R\leq z$ using the first part of the proof, but $y\geq z$ implies $y^R\nleq z$, so a contradiction. The fourth one is similar.
Aug 16, 2022 at 18:45 comment added Wouter Zandsteeg I think I understand the structure of the proof, but to get from the first four inequalities to the latter four, I can't see an easy way without using the theorem itself. I do understand the rest though.
Aug 16, 2022 at 18:02 comment added Apollo Thanks. A typical argument by contradiction in (structural) induction goes: suppose that the proposition does not hold. Then take a least counter-example X. In our case, this means all elements of X do satisfy the proposition. Then prove that this implies a contradiction, hence the proposition holds for X, and as X was a least counterexample, the proposition must hold generally. Here, he is proving $x+y\ngeq x+z$ implies $y\ngeq z$ (the converse direction) by contradiction.
Aug 16, 2022 at 17:20 history edited LSpice CC BY-SA 4.0
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Aug 16, 2022 at 16:25 comment added Wouter Zandsteeg I edited the question to make it more readable and clear what I don't understand
Aug 16, 2022 at 16:25 history edited Wouter Zandsteeg CC BY-SA 4.0
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Aug 16, 2022 at 15:47 comment added Apollo As far as the question goes, I think he's just doing a typical inductive argument by contradiction - choose a least putative counter-example (so that all subgames satisfy the induction hypothesis) and show it leads to a contradiction.
Aug 16, 2022 at 15:43 comment added Apollo It's much better if you actually type out the formulae into the question, rather than providing an ephemeral link to an image.
S Aug 16, 2022 at 15:25 review First questions
Aug 16, 2022 at 15:38
S Aug 16, 2022 at 15:25 history asked Wouter Zandsteeg CC BY-SA 4.0