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I want to prove correctness for an algorithm that is the composition of three functions.

In order to provide a readable description of the whole process, my document is structured top-down, i.e. it begins with a description of the composition followed by each part. Due to the size of the parts (the composition itself is rather trivial) the top-level view is on the same hierarchy as each part (a section of the chapter).

Because of this structure, I would like to discuss the correctness theorem in the beginning. Naturally, its proof is also rather trivial, given the (much more interesting) correctness of each part. These proofs I would like to present along each part (in a subsection). Therefore, the theorem depends on three lemmata, which are defined later in the document.

What is a good style to introduce these dependencies? Putting them before the theorem might seem a little bit "out of the blue", putting them after the theorem might disturb the flow of reading. Ideally, I'd like to introduce them on the fly in the proof of the theorem and present them later, but what is a good way to do so?

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In general, it is a good idea to give your reader a "roadmap" of the approach that you will take early on. You might even introduce the theorem itself early on, but leave the proof for later.

Once you have established your roadmap, then you can proceed in order, building up to the theorem that the reader knows is coming (referencing you plan in the narrative that connects the mathematics).

Finally, when you have all three lemmas in place, you can return to the theorem, restate it, and state the simple proof drawing on all three lemmas.

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