Add congruence instances for < and <= on semilattices

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

Lattice.agda

@@ -1,7 +1,7 @@
 module Lattice where
 
-open import Chain using (Chain; Height; done; step; concat)
 open import Equivalence
+open import Chain
 
 import Data.Nat.Properties as NatProps
 open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym)
@@ -35,10 +35,19 @@ record IsSemilattice {a} (A : Set a)
         ⊔-comm : (x y : A) → (x ⊔ y) ≈ (y ⊔ x)
         ⊔-idemp : (x : A) → (x ⊔ x) ≈ x
 
+    open IsEquivalence ≈-equiv public
+
     ≼-refl : ∀ (a : A) → a ≼ a
     ≼-refl a = (a , ⊔-idemp a)
 
-    open IsEquivalence ≈-equiv public
+    ≼-cong : ∀ {a₁ a₂ a₃ a₄ : A} → a₁ ≈ a₂ → a₃ ≈ a₄ → a₁ ≼ a₃ → a₂ ≼ a₄
+    ≼-cong a₁≈a₂ a₃≈a₄ (c₁ , a₁⊔c₁≈a₃) = (c₁ , ≈-trans (≈-⊔-cong (≈-sym a₁≈a₂) ≈-refl) (≈-trans a₁⊔c₁≈a₃ a₃≈a₄))
+
+    ≺-cong : ∀ {a₁ a₂ a₃ a₄ : A} → a₁ ≈ a₂ → a₃ ≈ a₄ → a₁ ≺ a₃ → a₂ ≺ a₄
+    ≺-cong a₁≈a₂ a₃≈a₄ (a₁≼a₃ , a₁̷≈a₃) =
+        ( ≼-cong a₁≈a₂ a₃≈a₄ a₁≼a₃
+        , λ a₂≈a₄ → a₁̷≈a₃ (≈-trans a₁≈a₂ (≈-trans a₂≈a₄ (≈-sym a₃≈a₄)))
+        )
 
 record IsLattice {a} (A : Set a)
     (_≈_ : A → A → Set a)
@@ -71,10 +80,11 @@ record IsFiniteHeightLattice {a} (A : Set a)
 
     field
         isLattice : IsLattice A _≈_ _⊔_ _⊓_
-        fixedHeight : Height (IsLattice._≺_ isLattice) h
+        fixedHeight : Chain.Height (IsLattice._≺_ isLattice) h
 
     open IsLattice isLattice public
 
+
 module _ {a b} {A : Set a} {B : Set b}
     (_≼₁_ : A → A → Set a) (_≼₂_ : B → B → Set b) where
 
@@ -86,8 +96,8 @@ module ChainMapping {a b} {A : Set a} {B : Set b}
     {_⊔₁_ : A → A → A} {_⊔₂_ : B → B → B}
     (slA : IsSemilattice A _≈₁_ _⊔₁_) (slB : IsSemilattice B _≈₂_ _⊔₂_) where
 
-    open IsSemilattice slA renaming (_≼_ to _≼₁_; _≺_ to _≺₁_)
-    open IsSemilattice slB renaming (_≼_ to _≼₂_; _≺_ to _≺₂_)
+    open IsSemilattice slA renaming (_≼_ to _≼₁_; _≺_ to _≺₁_; ≈-equiv to ≈₁-equiv)
+    open IsSemilattice slB renaming (_≼_ to _≼₂_; _≺_ to _≺₂_; ≈-equiv to ≈₂-equiv)
 
     Chain-map : ∀ (f : A → B) → Monotonic _≼₁_ _≼₂_ f → Injective _≈₁_ _≈₂_ f →
                 ∀ {a₁ a₂ : A} {n : ℕ} → Chain _≺₁_ a₁ a₂ n → Chain _≺₂_ (f a₁) (f a₂) n