Simplify AboveBelow a bit to avoid nested modules

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

Analysis/Sign.agda

@@ -42,22 +42,18 @@ open import Lattice.AboveBelow Sign _≡_ (record { ≈-refl = refl; ≈-sym = s
         ; ⊥ to ⊥ᵍ
         ; ⊤ to ⊤ᵍ
         ; [_] to [_]ᵍ
+        ; _≈_ to _≈ᵍ_
         ; ≈-⊥-⊥ to ≈ᵍ-⊥ᵍ-⊥ᵍ
         ; ≈-⊤-⊤ to ≈ᵍ-⊤ᵍ-⊤ᵍ
         ; ≈-lift to ≈ᵍ-lift
+        ; ≈-refl to ≈ᵍ-refl
         )
 -- 'sign' has no underlying lattice structure, so use the 'plain' above-below lattice.
-open AB.Plain using () renaming (finiteHeightLattice to finiteHeightLatticeᵍ-if-inhabited)
-
-finiteHeightLatticeᵍ = finiteHeightLatticeᵍ-if-inhabited 0ˢ
-
-open FiniteHeightLattice finiteHeightLatticeᵍ
-    using ()
+open AB.Plain 0ˢ using ()
     renaming
-        ( _≼_ to _≼ᵍ_
-        ; _≈_ to _≈ᵍ_
+        ( finiteHeightLattice to finiteHeightLatticeᵍ
+        ; _≼_ to _≼ᵍ_
         ; _⊔_ to _⊔ᵍ_
-        ; ≈-refl to ≈ᵍ-refl
         )
 
 plus : SignLattice → SignLattice → SignLattice

Lattice/AboveBelow.agda

@@ -68,7 +68,10 @@ data _≈_ : AboveBelow → AboveBelow → Set a where
 -- Any object can be wrapped in an 'above below' to make it a lattice,
 -- since ⊤ and ⊥ are the largest and least elements, and the rest are left
 -- unordered. That's what this module does.
-module Plain where
+--
+-- For convenience, ask for the underlying type to always be inhabited, to
+-- avoid requiring additional constraints in some of the proofs below.
+module Plain (x : A) where
     _⊔_ : AboveBelow → AboveBelow → AboveBelow
     ⊥ ⊔ x = x
     ⊤ ⊔ x = ⊤
@@ -296,7 +299,7 @@ module Plain where
         ; isLattice = isLattice
         }
 
-    open IsLattice isLattice using (_≼_; _≺_)
+    open IsLattice isLattice using (_≼_; _≺_) public
 
     ⊥≺[x] : ∀ (x : A) → ⊥ ≺ [ x ]
     ⊥≺[x] x = (≈-refl , λ ())
@@ -322,7 +325,6 @@ module Plain where
 
     open Chain _≈_ ≈-equiv (IsLattice._≺_ isLattice) (IsLattice.≺-cong isLattice)
 
-    module _ (x : A) where
     longestChain : Chain ⊥ ⊤ 2
     longestChain = step (⊥≺[x] x) ≈-refl (step ([x]≺⊤ x) ≈-⊤-⊤ (done ≈-⊤-⊤))
 

Main.agda

@@ -22,15 +22,13 @@ xyzw-Unique = push ((λ ()) ∷ (λ ()) ∷ (λ ()) ∷ []) (push ((λ ()) ∷ (
 open import Lattice using (IsFiniteHeightLattice; FiniteHeightLattice; Monotonic)
 
 open import Lattice.AboveBelow ⊤ _≡_ (record { ≈-refl = refl; ≈-sym = sym; ≈-trans = trans }) _≟ᵘ_  as AB using () renaming (≈-dec to ≈ᵘ-dec)
-open AB.Plain using () renaming (finiteHeightLattice to finiteHeightLatticeᵘ)
+open AB.Plain (Data.Unit.tt) using () renaming (finiteHeightLattice to fhlᵘ)
 
 showAboveBelow : AB.AboveBelow → String
 showAboveBelow AB.⊤ = "⊤"
 showAboveBelow AB.⊥ = "⊥"
 showAboveBelow (AB.[_] tt) = "()"
 
-fhlᵘ = finiteHeightLatticeᵘ (Data.Unit.tt)
-
 import Lattice.Bundles.FiniteValueMap
 open Lattice.Bundles.FiniteValueMap.FromFiniteHeightLattice String AB.AboveBelow _≟ˢ_ fhlᵘ xyzw-Unique ≈ᵘ-dec using (FiniteMap; ≈-dec) renaming (finiteHeightLattice to fhlⁱᵖ)