Add 'ExtendBelow' lattice, which adds new bottom to lattices

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

Lattice/ExtendBelow.agda

@@ -0,0 +1,169 @@
+open import Lattice
+
+module Lattice.ExtendBelow {a} (A : Set a)
+    {_≈₁_ : A → A → Set a} {_⊔₁_ : A → A → A} {_⊓₁_ : A → A → A}
+    {{lA : IsLattice A _≈₁_ _⊔₁_ _⊓₁_}} where
+
+open import Equivalence
+open import Showable using (Showable; show)
+open import Relation.Binary.Definitions using (Decidable)
+open import Relation.Binary.PropositionalEquality using (refl)
+open import Relation.Nullary using (Dec; ¬_; yes; no)
+
+open IsLattice lA using ()
+    renaming ( ≈-equiv to ≈₁-equiv
+             ; ≈-⊔-cong to ≈₁-⊔₁-cong
+             ; ⊔-assoc to ⊔₁-assoc; ⊔-comm to ⊔₁-comm; ⊔-idemp to ⊔₁-idemp
+             ; ≈-⊓-cong to ≈₁-⊓₁-cong
+             ; ⊓-assoc to ⊓₁-assoc; ⊓-comm to ⊓₁-comm; ⊓-idemp to ⊓₁-idemp
+             ; absorb-⊔-⊓ to absorb-⊔₁-⊓₁; absorb-⊓-⊔ to absorb-⊓₁-⊔₁
+             )
+
+open IsEquivalence ≈₁-equiv using ()
+    renaming (≈-refl to ≈₁-refl; ≈-sym to ≈₁-sym; ≈-trans to ≈₁-trans)
+
+data ExtendBelow : Set a where
+    [_] : A → ExtendBelow
+    ⊥ : ExtendBelow
+
+instance
+    showable : {{ showableA : Showable A }} → Showable ExtendBelow
+    showable = record
+        { show = (λ
+            { ⊥ → "⊥"
+            ; [ a ] → show a
+            })
+        }
+
+data _≈_ : ExtendBelow → ExtendBelow → Set a where
+    ≈-⊥-⊥ : ⊥ ≈ ⊥
+    ≈-lift : ∀ {x y : A} → x ≈₁ y → [ x ] ≈ [ y ]
+
+≈-refl : ∀ {ab : ExtendBelow} → ab ≈ ab
+≈-refl {⊥} = ≈-⊥-⊥
+≈-refl {[ x ]} = ≈-lift ≈₁-refl
+
+≈-sym : ∀ {ab₁ ab₂ : ExtendBelow} → ab₁ ≈ ab₂ → ab₂ ≈ ab₁
+≈-sym ≈-⊥-⊥ = ≈-⊥-⊥
+≈-sym (≈-lift x≈₁y) = ≈-lift (≈₁-sym x≈₁y)
+
+≈-trans : ∀ {ab₁ ab₂ ab₃ : ExtendBelow} → ab₁ ≈ ab₂ → ab₂ ≈ ab₃ → ab₁ ≈ ab₃
+≈-trans ≈-⊥-⊥ ≈-⊥-⊥ = ≈-⊥-⊥
+≈-trans (≈-lift a₁≈a₂) (≈-lift a₂≈a₃) = ≈-lift (≈₁-trans a₁≈a₂ a₂≈a₃)
+
+instance
+    ≈-equiv : IsEquivalence ExtendBelow _≈_
+    ≈-equiv = record
+        { ≈-refl = ≈-refl
+        ; ≈-sym = ≈-sym
+        ; ≈-trans = ≈-trans
+        }
+
+_⊔_ : ExtendBelow → ExtendBelow → ExtendBelow
+_⊔_ ⊥ x = x
+_⊔_ [ a₁ ] ⊥ = [ a₁ ]
+_⊔_ [ a₁ ] [ a₂ ] = [ a₁ ⊔₁ a₂ ]
+
+_⊓_ : ExtendBelow → ExtendBelow → ExtendBelow
+_⊓_ ⊥ x = ⊥
+_⊓_ [ _ ] ⊥ = ⊥
+_⊓_ [ a₁ ] [ a₂ ] = [ a₁ ⊓₁ a₂ ]
+
+≈-⊔-cong : ∀ {x₁ x₂ x₃ x₄} → x₁ ≈ x₂ → x₃ ≈ x₄ →
+           (x₁ ⊔ x₃) ≈ (x₂ ⊔ x₄)
+≈-⊔-cong .{⊥} .{⊥} {x₃} {x₄} ≈-⊥-⊥ [a₃]≈[a₄] = [a₃]≈[a₄]
+≈-⊔-cong {x₁ = [ a₁ ]} {x₂ = [ a₂ ]} .{⊥} .{⊥} [a₁]≈[a₂] ≈-⊥-⊥ = [a₁]≈[a₂]
+≈-⊔-cong {x₁ = [ a₁ ]} {x₂ = [ a₂ ]} {x₃ = [ a₃ ]} {x₄ = [ a₄ ]} (≈-lift a₁≈a₂) (≈-lift a₃≈a₄) = ≈-lift (≈₁-⊔₁-cong a₁≈a₂ a₃≈a₄)
+
+⊔-assoc : ∀ (x₁ x₂ x₃ : ExtendBelow) →  ((x₁ ⊔ x₂) ⊔ x₃) ≈ (x₁ ⊔ (x₂ ⊔ x₃))
+⊔-assoc ⊥ x₁ x₂ = ≈-refl
+⊔-assoc [ a₁ ] ⊥ x₂ = ≈-refl
+⊔-assoc [ a₁ ] [ a₂ ] ⊥ = ≈-refl
+⊔-assoc [ a₁ ] [ a₂ ] [ a₃ ] = ≈-lift (⊔₁-assoc a₁ a₂ a₃)
+
+⊔-comm : ∀ (x₁ x₂ : ExtendBelow) →  (x₁ ⊔ x₂) ≈ (x₂ ⊔ x₁)
+⊔-comm ⊥ ⊥ = ≈-refl
+⊔-comm ⊥ [ a₂ ] = ≈-refl
+⊔-comm [ a₁ ] ⊥ = ≈-refl
+⊔-comm [ a₁ ] [ a₂ ] = ≈-lift (⊔₁-comm a₁ a₂)
+
+⊔-idemp : ∀ (x : ExtendBelow) →  (x ⊔ x) ≈ x
+⊔-idemp ⊥ = ≈-refl
+⊔-idemp [ a ] = ≈-lift (⊔₁-idemp a)
+
+≈-⊓-cong : ∀ {x₁ x₂ x₃ x₄} → x₁ ≈ x₂ → x₃ ≈ x₄ →
+           (x₁ ⊓ x₃) ≈ (x₂ ⊓ x₄)
+≈-⊓-cong .{⊥} .{⊥} {x₃} {x₄} ≈-⊥-⊥ [a₃]≈[a₄] = ≈-⊥-⊥
+≈-⊓-cong {x₁ = [ a₁ ]} {x₂ = [ a₂ ]} .{⊥} .{⊥} [a₁]≈[a₂] ≈-⊥-⊥ = ≈-⊥-⊥
+≈-⊓-cong {x₁ = [ a₁ ]} {x₂ = [ a₂ ]} {x₃ = [ a₃ ]} {x₄ = [ a₄ ]} (≈-lift a₁≈a₂) (≈-lift a₃≈a₄) = ≈-lift (≈₁-⊓₁-cong a₁≈a₂ a₃≈a₄)
+
+⊓-assoc : ∀ (x₁ x₂ x₃ : ExtendBelow) →  ((x₁ ⊓ x₂) ⊓ x₃) ≈ (x₁ ⊓ (x₂ ⊓ x₃))
+⊓-assoc ⊥ x₁ x₂ = ≈-refl
+⊓-assoc [ a₁ ] ⊥ x₂ = ≈-refl
+⊓-assoc [ a₁ ] [ a₂ ] ⊥ = ≈-refl
+⊓-assoc [ a₁ ] [ a₂ ] [ a₃ ] = ≈-lift (⊓₁-assoc a₁ a₂ a₃)
+
+⊓-comm : ∀ (x₁ x₂ : ExtendBelow) →  (x₁ ⊓ x₂) ≈ (x₂ ⊓ x₁)
+⊓-comm ⊥ ⊥ = ≈-refl
+⊓-comm ⊥ [ a₂ ] = ≈-refl
+⊓-comm [ a₁ ] ⊥ = ≈-refl
+⊓-comm [ a₁ ] [ a₂ ] = ≈-lift (⊓₁-comm a₁ a₂)
+
+⊓-idemp : ∀ (x : ExtendBelow) →  (x ⊓ x) ≈ x
+⊓-idemp ⊥ = ≈-refl
+⊓-idemp [ a ] = ≈-lift (⊓₁-idemp a)
+
+absorb-⊔-⊓ : (x₁ x₂ : ExtendBelow) → (x₁ ⊔ (x₁ ⊓ x₂)) ≈ x₁
+absorb-⊔-⊓ ⊥ ⊥ = ≈-refl
+absorb-⊔-⊓ ⊥ [ a₂ ] = ≈-refl
+absorb-⊔-⊓ [ a₁ ] ⊥ = ≈-refl
+absorb-⊔-⊓ [ a₁ ] [ a₂ ] = ≈-lift (absorb-⊔₁-⊓₁ a₁ a₂)
+
+absorb-⊓-⊔ : (x₁ x₂ : ExtendBelow) → (x₁ ⊓ (x₁ ⊔ x₂)) ≈ x₁
+absorb-⊓-⊔ ⊥ ⊥ = ≈-refl
+absorb-⊓-⊔ ⊥ [ a₂ ] = ≈-refl
+absorb-⊓-⊔ [ a₁ ] ⊥ = ⊓-idemp [ a₁ ]
+absorb-⊓-⊔ [ a₁ ] [ a₂ ] = ≈-lift (absorb-⊓₁-⊔₁ a₁ a₂)
+
+instance
+    isJoinSemilattice : IsSemilattice ExtendBelow _≈_ _⊔_
+    isJoinSemilattice = record
+        { ≈-equiv = ≈-equiv
+        ; ≈-⊔-cong = ≈-⊔-cong
+        ; ⊔-assoc = ⊔-assoc
+        ; ⊔-comm = ⊔-comm
+        ; ⊔-idemp = ⊔-idemp
+        }
+
+    isMeetSemilattice : IsSemilattice ExtendBelow _≈_ _⊓_
+    isMeetSemilattice = record
+        { ≈-equiv = ≈-equiv
+        ; ≈-⊔-cong = ≈-⊓-cong
+        ; ⊔-assoc = ⊓-assoc
+        ; ⊔-comm = ⊓-comm
+        ; ⊔-idemp = ⊓-idemp
+        }
+
+    isLattice : IsLattice ExtendBelow _≈_ _⊔_ _⊓_
+    isLattice = record
+        { joinSemilattice = isJoinSemilattice
+        ; meetSemilattice = isMeetSemilattice
+        ; absorb-⊔-⊓ = absorb-⊔-⊓
+        ; absorb-⊓-⊔ = absorb-⊓-⊔
+        }
+
+module _ {{≈₁-Decidable : IsDecidable _≈₁_}} where
+    open IsDecidable ≈₁-Decidable using () renaming (R-dec to ≈₁-dec)
+
+    ≈-dec : Decidable _≈_
+    ≈-dec ⊥ ⊥ = yes ≈-⊥-⊥
+    ≈-dec [ a₁ ] [ a₂ ]
+        with ≈₁-dec a₁ a₂
+    ...   | yes a₁≈a₂ = yes (≈-lift a₁≈a₂)
+    ...   | no a₁̷≈a₂ = no (λ { (≈-lift a₁≈a₂) → a₁̷≈a₂ a₁≈a₂ })
+    ≈-dec ⊥ [ _ ] = no (λ ())
+    ≈-dec [ _ ] ⊥ = no (λ ())
+
+    instance
+        ≈-Decidable : IsDecidable _≈_
+        ≈-Decidable = record { R-dec = ≈-dec }