Fold Isomorphism module into Lattice.lean

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

lean/Spa/Lattice.lean

@@ -107,6 +107,25 @@ lemma le_top (a : α) : a ≤ (⊤ : α) := by
     rw [RelSeries.snoc_length] at hbound
     omega
 
+/-- This is something like a lemma about isomorphic types having the same height.
+    Given a finite-height lattice `α`, lattice `β`, and a `Monotone` bijection
+    between the two, we can show that lattice `β` also has a finite height.
+
+    The proof is fairly trivial: the longest chain in `α` can be transported
+    to be a longest chain in `β` (by monotonicity), establishing a height witness.
+    At the same time, any chain in `β` can be transported to a chain in `α`,
+    and must be bounded by the same height by `FiniteHeightLattice.chains_bounded`. -/
+def transport {α β : Type*} [Lattice β]
+    [I : FiniteHeightLattice α] (f : α → β) (g : β → α)
+    (hf : Monotone f) (hg : Monotone g)
+    (hgf : Function.LeftInverse g f) (hfg : Function.LeftInverse f g) :
+    FiniteHeightLattice β where
+  toLattice := inferInstance
+  longestChain :=
+    I.longestChain.map f (hf.strictMono_of_injective hgf.injective)
+  chains_bounded := fun c =>
+    I.chains_bounded (c.map g (hg.strictMono_of_injective hfg.injective))
+
 end FiniteHeightLattice
 
 end Spa