Require bottom element to actually be bottom; finish proof

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

Lattice.agda

@@ -260,6 +260,12 @@ record IsFiniteHeightLattice {a} (A : Set a)
 
     open MyChain.Height fixedHeight using (⊥; ⊤) public
 
+    Known-⊥ : Set a
+    Known-⊥ = ∀ (a : A) → ⊥ ≼ a
+
+    Known-⊤ : Set a
+    Known-⊤ = ∀ (a : A) → a ≼ ⊤
+
     -- If the equality is decidable, we can prove that the top and bottom
     -- elements of the chain are least and greatest elements of the lattice
     module _ {{≈-Decidable : IsDecidable _≈_}} where
@@ -267,7 +273,7 @@ record IsFiniteHeightLattice {a} (A : Set a)
 
         open MyChain.Height fixedHeight using (bounded; longestChain)
 
-        ⊥≼ : ∀ (a : A) → ⊥ ≼ a
+        ⊥≼ : Known-⊥
         ⊥≼ a with ≈-dec a ⊥
         ...   | yes a≈⊥ = ≼-cong a≈⊥ ≈-refl (≼-refl a)
         ...   | no a̷≈⊥ with ≈-dec ⊥ (a ⊓ ⊥)