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I would like to know how to reorder goals in the following situation:

lemma "P=Q"
  proof (rule iffI, (*here I would like to swap goal order*), rule ccontr)
oops

I would like a solution that doesn't involve changing the lemma statement. I realise that prefer and defer can be used in apply-style proofs, but I would like to have a method that can be used in the proof (...) part.

Edit:

As Andreas Lochbihler says, writing rule iffI[rotated] works in the above example. However, is it possible to swap the goal order in the following situation without changing the statement of the lemma?

lemma "P==>Q" "Q==>P"
  proof ((*here I would like to swap goal order*), rule ccontr)
oops

This example may seem contrived, but I feel that there may be situations where it is inconvenient to change the statement of the lemma, or it is necessary to swap goal order when there is no previous application of a rule such as iffI.

IIM
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1 Answers1

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The order of the subgoals is determined by the order of the assumptions of the rule you apply. So it suffices to swap the assumptions of the iffI rule, e.g., using the attribute [rotated] as in

proof(rule iffI[rotated], rule ccontr)

In general, there is no proof method to change the order of the goals. And if you are thinking about using this with more sophisticated proof automation like auto, I'd heavily advise you against doing these kinds of things. Proof scripts with lots of automation in it should work independently of the order of the goals. Otherwise, your proofs will break easily when something in the proof automation setup changes.

However, a few low-level proof tactics allow to use explicit goal addressing (mostly those that end on _tac). For example,

proof(rule iffI, rule_tac [2] ccontr)

applies the ccontr rule to the second subgoal instead of the first.

Andreas Lochbihler
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  • This is very helpful. However, do you know a more general way: I will add an example to my question which should explain why `rotated` might not be general enough. – IIM Sep 20 '17 at 09:46
  • I've changed my answer accordingly. There is proof method to reorder subgoals within a method expression. In principle, you could implement your own proof method that does that. – Andreas Lochbihler Sep 21 '17 at 06:21