Prove that 'first' presrves equality

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

Lattice/FiniteValueMap.agda

@@ -16,20 +16,23 @@ module Lattice.FiniteValueMap (A : Set) (B : Set)
     (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
 
 open import Data.List using (List; length; []; _∷_)
-open import Utils using (Unique; push; empty)
+open import Data.Product using (Σ; proj₁; proj₂; _×_)
+open import Data.Empty using (⊥-elim)
+open import Utils using (Unique; push; empty; All¬-¬Any)
 open import Data.Product using (_,_)
 open import Data.List.Properties using (∷-injectiveʳ)
 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 Lattice.Map A B _≈₂_ _⊔₂_ _⊓₂_ ≈-dec-A lB using (subset-impl)
+open import Lattice.Map A B _≈₂_ _⊔₂_ _⊓₂_ ≈-dec-A lB using (subset-impl; locate; forget; _∈_; Map-functional)
 open import Lattice.FiniteMap A B _≈₂_ _⊔₂_ _⊓₂_ ≈-dec-A lB public
 
 module IterProdIsomorphism where
     open import Data.Unit using (⊤; tt)
-    open import Lattice.Unit using () renaming (_≈_ to _≈ᵘ_; _⊔_ to _⊔ᵘ_; _⊓_ to _⊓ᵘ_; ≈-dec to ≈ᵘ-dec; isLattice to isLatticeᵘ)
+    open import Lattice.Unit using () renaming (_≈_ to _≈ᵘ_; _⊔_ to _⊔ᵘ_; _⊓_ to _⊓ᵘ_; ≈-dec to ≈ᵘ-dec; isLattice to isLatticeᵘ; ≈-equiv to ≈ᵘ-equiv)
     open import Lattice.IterProd _≈₂_ _≈ᵘ_ _⊔₂_ _⊔ᵘ_ _⊓₂_ _⊓ᵘ_ lB isLatticeᵘ as IP using (IterProd)
+    open IsLattice lB using () renaming (≈-trans to ≈₂-trans; ≈-sym to ≈₂-sym)
 
     from : ∀ {ks : List A} → FiniteMap ks → IterProd (length ks)
     from {[]} (([] , _) , _) = tt
@@ -54,6 +57,9 @@ module IterProdIsomorphism where
         _≈ᵐ_ : ∀ {ks : List A} → FiniteMap ks → FiniteMap ks → Set
         _≈ᵐ_ {ks} = _≈_ ks
 
+        _⊆ᵐ_ : ∀ {ks₁ ks₂ : List A} → FiniteMap ks₁ → FiniteMap ks₂ → Set
+        _⊆ᵐ_ fm₁ fm₂ = subset-impl (proj₁ (proj₁ fm₁)) (proj₁ (proj₁ fm₂))
+
         _≈ⁱᵖ_ : ∀ {ks : List A} → IterProd (length ks) → IterProd (length ks) → Set
         _≈ⁱᵖ_ {ks} = IP._≈_ (length ks)
 
@@ -91,3 +97,42 @@ module IterProdIsomorphism where
                     m₂⊆m₁ k' v' (there k',v'∈kvs'₂) =
                         let (v'' , (v'≈v'' , k',v''∈kvs'₁)) = kvs'₂⊆kvs'₁ k' v' k',v'∈kvs'₂
                         in (v'' , (v'≈v'' , there k',v''∈kvs'₁))
+
+    private
+        first-key-in-map : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → Σ B (λ v → (k , v) ∈ proj₁ fm)
+        first-key-in-map (((k , v) ∷ _ , _) , refl) = (v , here refl)
+
+        from-first-value : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → proj₁ (from fm) ≈₂ proj₁ (first-key-in-map fm)
+        from-first-value {k} {ks} (((k , v) ∷ _ , push _ _) , refl) = IsLattice.≈-refl lB
+
+        pop : ∀ {k : A} {ks : List A} → FiniteMap (k ∷ ks) → FiniteMap ks
+        pop (((_ ∷ kvs') , push _ ukvs') , refl) = ((kvs' , ukvs') , refl)
+
+        pop-≈ :  ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) → fm₁ ≈ᵐ fm₂ → pop fm₁ ≈ᵐ pop fm₂
+        pop-≈ {k} {ks} fm₁ fm₂ (fm₁⊆fm₂ , fm₂⊆fm₁) = (narrow fm₁⊆fm₂ , narrow fm₂⊆fm₁)
+            where
+                narrow₁ : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} → fm₁ ⊆ᵐ fm₂  → pop fm₁ ⊆ᵐ fm₂
+                narrow₁ {(_ ∷ _ , push _ _) , refl} kvs₁⊆kvs₂ k' v' k',v'∈kvs'₁ = kvs₁⊆kvs₂ k' v' (there k',v'∈kvs'₁)
+
+                narrow₂ : ∀ {fm₁ : FiniteMap ks} {fm₂ : FiniteMap (k ∷ ks)} → fm₁ ⊆ᵐ fm₂  → fm₁ ⊆ᵐ pop fm₂
+                narrow₂ {fm₁} {fm₂ = (_ ∷ kvs'₂ , push k≢ks _) , kvs≡ks@refl} kvs₁⊆kvs₂ k' v' k',v'∈kvs'₁
+                    with kvs₁⊆kvs₂ k' v' k',v'∈kvs'₁
+                ...   | (v'' , (v'≈v'' , here refl)) rewrite sym (proj₂ fm₁) = ⊥-elim (All¬-¬Any k≢ks (forget {m = proj₁ fm₁} k',v'∈kvs'₁))
+                ...   | (v'' , (v'≈v'' , there k',v'∈kvs'₂)) = (v'' , (v'≈v'' , k',v'∈kvs'₂))
+
+                narrow : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} → fm₁ ⊆ᵐ fm₂  → pop fm₁ ⊆ᵐ pop fm₂
+                narrow {fm₁} {fm₂} x = narrow₂ {pop fm₁} (narrow₁ {fm₂ = fm₂} x)
+
+        from-rest : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → proj₂ (from fm) ≡ from (pop fm)
+        from-rest (((_ ∷ kvs') , push _ ukvs') , refl) = refl
+
+    from-preserves-≈ : ∀ {ks : List A} → (fm₁ fm₂ : FiniteMap ks) → fm₁ ≈ᵐ fm₂ → (_≈ⁱᵖ_ {ks}) (from fm₁) (from fm₂)
+    from-preserves-≈ {[]} (([] , _) , _) (([] , _) , _) _ = IsEquivalence.≈-refl ≈ᵘ-equiv
+    from-preserves-≈ {k ∷ ks'} fm₁@(m₁ , _) fm₂@(m₂ , _) fm₁≈fm₂@(kvs₁⊆kvs₂ , kvs₂⊆kvs₁)
+        with first-key-in-map fm₁ | first-key-in-map fm₂ | from-first-value fm₁ | from-first-value fm₂
+    ...   | (v₁ , k,v₁∈fm₁) | (v₂ , k,v₂∈fm₂) | fv₁≈v₁ | fv₂≈v₂
+            with kvs₁⊆kvs₂ _ _ k,v₁∈fm₁
+    ...       | (v₁' , (v₁≈v₁' , k,v₁'∈fm₂))
+                rewrite Map-functional {m = m₂} k,v₂∈fm₂ k,v₁'∈fm₂
+                rewrite from-rest fm₁ rewrite from-rest fm₂
+                = (≈₂-trans fv₁≈v₁ (≈₂-trans v₁≈v₁' (≈₂-sym fv₂≈v₂)) , from-preserves-≈ (pop fm₁) (pop fm₂) (pop-≈ fm₁ fm₂ fm₁≈fm₂))