Strengthen lemma about IterProd bottom to definition equality

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

Lattice/IterProd.agda

@@ -13,6 +13,7 @@ module Lattice.IterProd {a} {A B : Set a}
 open import Agda.Primitive using (lsuc)
 open import Data.Nat using (ℕ; zero; suc; _+_)
 open import Data.Product using (_×_; _,_; proj₁; proj₂)
+open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; cong)
 open import Utils using (iterate)
 open import Chain using (Height)
 
@@ -59,7 +60,7 @@ private
             fixedHeight : IsLattice.FixedHeight isLattice height
             ≈-dec : IsDecidable _≈_
 
-            ⊥-correct : Height.⊥ fixedHeight ≈ ⊥
+            ⊥-correct : Height.⊥ fixedHeight ≡ ⊥
 
     record Everything (k : ℕ) : Set (lsuc a) where
         T = IterProd k
@@ -84,7 +85,7 @@ private
             { height = RequiredForFixedHeight.h₂ req
             ; fixedHeight = RequiredForFixedHeight.fhB req
             ; ≈-dec = RequiredForFixedHeight.≈₂-dec req
-            ; ⊥-correct = IsLattice.≈-refl lB
+            ; ⊥-correct = refl
             }
         }
     everything (suc k') = record
@@ -104,7 +105,9 @@ private
                         (RequiredForFixedHeight.h₁ req) (IsFiniteHeightWithBotAndDecEq.height fhlRest)
                         (RequiredForFixedHeight.fhA req) (IsFiniteHeightWithBotAndDecEq.fixedHeight fhlRest)
                     ; ≈-dec = P.≈-dec (RequiredForFixedHeight.≈₁-dec req) (IsFiniteHeightWithBotAndDecEq.≈-dec fhlRest)
-                    ; ⊥-correct = (IsLattice.≈-refl lA , IsFiniteHeightWithBotAndDecEq.⊥-correct fhlRest)
+                    ; ⊥-correct =
+                        cong ((Height.⊥ (RequiredForFixedHeight.fhA req)) ,_)
+                             (IsFiniteHeightWithBotAndDecEq.⊥-correct fhlRest)
                     }
         }
         where
@@ -158,3 +161,7 @@ module _ (k : ℕ) where
             ; _⊓_ = _⊓_
             ; isFiniteHeightLattice = isFiniteHeightLattice
             }
+
+        ⊥-built : Height.⊥ fixedHeight ≡ (build (Height.⊥ fhA) (Height.⊥ fhB) k)
+        ⊥-built = IsFiniteHeightWithBotAndDecEq.⊥-correct (Everything.isFiniteHeightIfSupported (everything k) required)
+