Prove that the bottom map's valyes are all bottoms

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

Lattice/FiniteValueMap.agda

@@ -28,6 +28,7 @@ open import Data.List.Relation.Unary.All using (All)
 open import Data.List.Relation.Unary.Any using (Any; here; there)
 open import Relation.Nullary using (¬_)
 open import Isomorphism using (IsInverseˡ; IsInverseʳ)
+open import Chain using (Height)
 
 open import Lattice.Map ≡-dec-A lB
     using
@@ -104,6 +105,14 @@ module IterProdIsomorphism where
         _∈ᵐ_ : ∀ {ks : List A} → A × B → FiniteMap ks → Set
         _∈ᵐ_ {ks} = _∈_ ks
 
+    to-build : ∀ {b : B} {ks : List A} (uks : Unique ks) →
+               let fm = to uks (IP.build b tt (length ks))
+               in ∀ (k : A) (v : B) → (k , v) ∈ᵐ fm → v ≡ b
+    to-build {b} {k ∷ ks'} (push _ uks') k v (here refl) = refl
+    to-build {b} {k ∷ ks'} (push _ uks') k' v (there k',v∈m') =
+        to-build {ks = ks'} uks' k' v k',v∈m'
+
+
     -- The left inverse is: from (to x) = x
     from-to-inverseˡ : ∀ {ks : List A} (uks : Unique ks) →
                       IsInverseˡ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {length ks})
@@ -407,3 +416,12 @@ module IterProdIsomorphism where
             (to-⊔-distr uks) (from-⊔-distr {ks})
             (from-to-inverseʳ uks) (from-to-inverseˡ uks)
             using (isFiniteHeightLattice; finiteHeightLattice) public
+
+        -- Helpful lemma: all entries of the 'bottom' map are assigned to bottom.
+
+        open Height (IsFiniteHeightLattice.fixedHeight isFiniteHeightLattice) using (⊥)
+
+        ⊥-contains-bottoms : ∀ {k : A} {v : B} → (k , v) ∈ᵐ ⊥ → v ≡ (Height.⊥ fhB)
+        ⊥-contains-bottoms {k} {v} k,v∈⊥
+            rewrite IP.⊥-built (length ks) ≈₂-dec ≈ᵘ-dec h₂ 0 fhB fixedHeightᵘ =
+            to-build uks k v k,v∈⊥