Move proof of least element into FiniteHeightLattice

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

Lattice.agda

@@ -4,7 +4,8 @@ open import Equivalence
 import Chain
 
 open import Relation.Binary.Core using (_Preserves_⟶_ )
-open import Relation.Nullary using (Dec; ¬_)
+open import Relation.Nullary using (Dec; ¬_; yes; no)
+open import Data.Empty using (⊥-elim)
 open import Data.Nat as Nat using (ℕ)
 open import Data.Product using (_×_; Σ; _,_)
 open import Data.Sum using (_⊎_; inj₁; inj₂)
@@ -236,6 +237,28 @@ record IsFiniteHeightLattice {a} (A : Set a)
     field
         {{fixedHeight}} : FixedHeight h
 
+    -- 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
+        open IsDecidable ≈-Decidable using () renaming (R-dec to ≈-dec)
+
+        module MyChain = Chain _≈_ ≈-equiv _≺_ ≺-cong
+        open MyChain.Height fixedHeight using (⊥; ⊤) public
+        open MyChain.Height fixedHeight using (bounded; longestChain)
+
+        ⊥≼ : ∀ (a : A) → ⊥ ≼ a
+        ⊥≼ a with ≈-dec a ⊥
+        ...   | yes a≈⊥ = ≼-cong a≈⊥ ≈-refl (≼-refl a)
+        ...   | no a̷≈⊥ with ≈-dec ⊥ (a ⊓ ⊥)
+        ...         | yes ⊥≈a⊓⊥ = ≈-trans (⊔-comm ⊥ a) (≈-trans (≈-⊔-cong (≈-refl {a}) ⊥≈a⊓⊥) (absorb-⊔-⊓ a ⊥))
+        ...         | no ⊥ᴬ̷≈a⊓⊥ = ⊥-elim (MyChain.Bounded-suc-n bounded (MyChain.step x≺⊥ ≈-refl longestChain))
+                        where
+                            ⊥⊓a̷≈⊥ : ¬ (⊥ ⊓ a) ≈ ⊥
+                            ⊥⊓a̷≈⊥ = λ ⊥⊓a≈⊥ → ⊥ᴬ̷≈a⊓⊥ (≈-trans (≈-sym ⊥⊓a≈⊥) (⊓-comm _ _))
+
+                            x≺⊥ : (⊥ ⊓ a) ≺ ⊥
+                            x≺⊥ = (≈-trans (⊔-comm _ _) (≈-trans (≈-refl {⊥ ⊔ (⊥ ⊓ a)}) (absorb-⊔-⊓ ⊥ a)) , ⊥⊓a̷≈⊥)
+
 module ChainMapping {a b} {A : Set a} {B : Set b}
     {_≈₁_ : A → A → Set a} {_≈₂_ : B → B → Set b}
     {_⊔₁_ : A → A → A} {_⊔₂_ : B → B → B}