45 lines
1.0 KiB
Lean4
45 lines
1.0 KiB
Lean4
import Spa.Lattice.Prod
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import Spa.Lattice.Unit
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/-!
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# Iterated Products
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Given two types $\alpha$ and $\beta$ and a number $n$, produces
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an iterated product:
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$$
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\overbrace{\alpha \times \ldots \times \alpha}^{n\ \text{times}} × \beta
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$$
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This is mostly a stepping stone for isomorphisms. In
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`Spa/Lattice/Prod.lean`, By decomposing types such as `Fin n → α` into
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`IterProd α PUnit n`, we can automatically get a proof of their finite
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height via `Spa.FiniteHeightLattice.transport`.
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-/
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namespace Spa
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universe u
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def IterProd (A B : Type u) : ℕ → Type u
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| 0 => B
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| k + 1 => A × IterProd A B k
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namespace IterProd
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variable {A B : Type u}
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def fixedHeight [FiniteHeightLattice A] [FiniteHeightLattice B] :
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∀ k, FiniteHeightLattice (IterProd A B k)
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| 0 => inferInstanceAs (FiniteHeightLattice B)
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| k + 1 => @Spa.prod A (IterProd A B k) _ (fixedHeight k)
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instance finiteHeight [FiniteHeightLattice A] [FiniteHeightLattice B] (k : ℕ) :
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FiniteHeightLattice (IterProd A B k) := fixedHeight k
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end IterProd
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end Spa
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