Write more documentation

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

lean/Spa/Lattice/Prod.lean

@@ -1,11 +1,46 @@
 import Spa.Lattice
 
+/-!
+
+# Product Lattice
+
+This file provides a proof that, in addition to being a lattice,
+the product of two types $\alpha \times \beta$ forms a `Spa.FiniteHeightLattice`
+if both $\alpha$ and $\beta$ have a finite height.
+
+The proof proceeds by "unzipping" a chain:
+
+$$
+(a_1, b_1) < (a_1, b_2) < \ldots < (a_n, b_m)
+$$
+
+In which, at each step, either an $\alpha$ or $\beta$ element
+might ratchet up, into two chains:
+
+$$
+\begin{aligned}
+a_1 < \ldots < a_n \\
+b_1 < \ldots < b_m
+\end{aligned}
+$$
+
+Because at least one of the two "unzipped" chains grows with
+each element of the product chain, the full chain length
+can't exceed the sum of the two components. By the definition
+of finite height, these two chains are bounded, and therefore,
+the product chain is bounded too.
+
+-/
+
 namespace Spa
 
 section Unzip
 
 variable {α β : Type*} [PartialOrder α] [PartialOrder β]
 
+/-- The unzipping lemma: any chain (`LTSeries`) of product
+    elements can be decomposed into chains of components,
+    whose lengths bound the chain. -/
 lemma LTSeries.exists_unzip (c : LTSeries (α × β)) :
     ∃ (c₁ : LTSeries α) (c₂ : LTSeries β),
       c₁.head = c.head.1 ∧ c₁.last = c.last.1 ∧
@@ -60,6 +95,9 @@ section FixedHeight
 
 variable {α β : Type*}
 
+/-- The longest possible chain is one in which only one of the components grows
+    at a time, making the maximum height of $\alpha \times \beta$ be
+    $\text{height}_\alpha + \text{height}_\beta$. -/
 instance prod [A : FiniteHeightLattice α] [B : FiniteHeightLattice β] :
     FiniteHeightLattice (α × β) where
   toLattice := inferInstance