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POIsabelle: Use of the "induct" or "induct_tac" methods
 

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  1. POIsabelle: Use of the "induct" or "induct_tac" methods
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    <p>Let's say I have a lemma about a simple inductively-defined set:</p> <pre><code>inductive_set foo :: "'a ⇒ 'a list set" for x :: 'a where "[] ∈ foo x" | "[x] ∈ foo x" lemma "⋀x y. y ∈ foo x ⟹ qux x y ⟹ baz x y" </code></pre> <p>(It's important to me that the "⋀x y" bit stays, because the lemma is actually stating the state of my proof in the middle of a long apply chain.)</p> <p>I'm having trouble starting the proof of this lemma. I would like to proceed by rule induction. </p> <h2>First attempt</h2> <p>I tried writing</p> <pre><code>apply (induct rule: foo.induct) </code></pre> <p>but that doesn't work: the <code>induct</code> method fails. I find I can get around this by fixing <code>x</code> and <code>y</code> explicitly, and then invoking the <code>induct</code> method, like so:</p> <pre><code>proof - fix x :: 'a fix y :: "'a list" assume "y ∈ foo x" and "qux x y" thus "baz x y" apply (induct rule: foo.induct) oops </code></pre> <p>However, since I'm actually in the middle of an apply chain, I would rather not enter a structured proof block. </p> <h2>Second attempt</h2> <p>I tried using the <code>induct_tac</code> method instead, but unfortunately <code>induct_tac</code> does not apply the <code>foo.induct</code> rule in the way I would like. If I type</p> <pre><code>apply (induct_tac rule: foo.induct, assumption) </code></pre> <p>then the first subgoal is</p> <pre><code>⋀x y. y ∈ foo x ⟹ qux x y ⟹ baz x [] </code></pre> <p>which is not what I want: I wanted <code>qux x []</code> instead of <code>qux x y</code>. The <code>induct</code> method got this right, but had other problems, discussed above.</p>
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