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Add an instance of Semilattice for Map.

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2023-07-30 20:36:19 -07:00
parent a4eaa6269f
commit 1b7a3f02eb
2 changed files with 123 additions and 93 deletions
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28
Lattice.agda
View File

@ -216,6 +216,34 @@ module IsSemilatticeInstances where
; ⊔-idemp = ⊔-idemp ; ⊔-idemp = ⊔-idemp
} }
module ForMap {a} {A B : Set a}
(≡-dec-A : Decidable (_≡_ {a} {A}))
(_≈₂_ : B → B → Set a)
(_⊔₂_ : B → B → B)
(sB : IsSemilattice B _≈₂_ _⊔₂_) where
open import Map A B ≡-dec-A
private
infix 4 _≈_
infixl 20 _⊔_
_≈_ : Map → Map → Set a
_≈_ = lift (_≈₂_)
_⊔_ : Map → Map → Map
m₁ ⊔ m₂ = union _⊔₂_ m₁ m₂
module MapEquiv = IsEquivalenceInstances.ForMap A B ≡-dec-A _≈₂_ (IsSemilattice.≈-equiv sB)
MapIsUnionSemilattice : IsSemilattice Map _≈_ _⊔_
MapIsUnionSemilattice = record
{ ≈-equiv = MapEquiv.LiftEquivalence
; ⊔-assoc = union-assoc _≈₂_ (IsSemilattice.≈-refl sB) (IsSemilattice.≈-sym sB) _⊔₂_ (IsSemilattice.⊔-assoc sB)
; ⊔-comm = union-comm _≈₂_ (IsSemilattice.≈-refl sB) (IsSemilattice.≈-sym sB) _⊔₂_ (IsSemilattice.⊔-comm sB)
; ⊔-idemp = union-idemp _≈₂_ (IsSemilattice.≈-refl sB) (IsSemilattice.≈-sym sB) _⊔₂_ (IsSemilattice.⊔-idemp sB)
}
module IsLatticeInstances where module IsLatticeInstances where
module ForNat where module ForNat where
open Nat open Nat

58
Map.agda
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@ -342,14 +342,16 @@ module _ (_≈_ : B → B → Set b) where
lift : Map → Map → Set (a ⊔ b) lift : Map → Map → Set (a ⊔ b)
lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁ lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁
module _ (f : B → B → B) where module _ (≈-refl : ∀ {b : B} → b ≈ b)
(≈-sym : ∀ {b₁ b₂ : B} → b₁ ≈ b₂ → b₂ ≈ b₁)
(f : B → B → B) where
module I = ImplInsert f module I = ImplInsert f
module _ (f-idemp : ∀ (b : B) → f b b ≡ b) where module _ (f-idemp : ∀ (b : B) → f b b ≈ b) where
union-idemp : ∀ (m : Map) → lift (_≡_) (union f m m) m union-idemp : ∀ (m : Map) → lift (union f m m) m
union-idemp m@(l , u) = (mm-m-subset , m-mm-subset) union-idemp m@(l , u) = (mm-m-subset , m-mm-subset)
where where
mm-m-subset : subset (_≡_) (union f m m) m mm-m-subset : subset (union f m m) m
mm-m-subset k v k,v∈mm mm-m-subset k v k,v∈mm
with Expr-Provenance f k ((` m) ∪ (` m)) (∈-cong proj₁ k,v∈mm) with Expr-Provenance f k ((` m) ∪ (` m)) (∈-cong proj₁ k,v∈mm)
... | (_ , (bothᵘ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm)) ... | (_ , (bothᵘ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm))
@ -359,14 +361,14 @@ module _ (f : B → B → B) where
... | (_ , (in₁ (single {v'} v'∈m) k∉km , _)) = absurd (k∉km (∈-cong proj₁ v'∈m)) ... | (_ , (in₁ (single {v'} v'∈m) k∉km , _)) = absurd (k∉km (∈-cong proj₁ v'∈m))
... | (_ , (in₂ k∉km (single {v''} v''∈m) , _)) = absurd (k∉km (∈-cong proj₁ v''∈m)) ... | (_ , (in₂ k∉km (single {v''} v''∈m) , _)) = absurd (k∉km (∈-cong proj₁ v''∈m))
m-mm-subset : subset (_≡_) m (union f m m) m-mm-subset : subset m (union f m m)
m-mm-subset k v k,v∈m = (f v v , (sym (f-idemp v) , I.union-combines u u k,v∈m k,v∈m)) m-mm-subset k v k,v∈m = (f v v , (≈-sym (f-idemp v) , I.union-combines u u k,v∈m k,v∈m))
module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≡ f b₂ b₁) where module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≈ f b₂ b₁) where
union-comm : ∀ (m₁ m₂ : Map) → lift (_≡_) (union f m₁ m₂) (union f m₂ m₁) union-comm : ∀ (m₁ m₂ : Map) → lift (union f m₁ m₂) (union f m₂ m₁)
union-comm m₁ m₂ = (union-comm-subset m₁ m₂ , union-comm-subset m₂ m₁) union-comm m₁ m₂ = (union-comm-subset m₁ m₂ , union-comm-subset m₂ m₁)
where where
union-comm-subset : ∀ (m₁ m₂ : Map) → subset (_≡_) (union f m₁ m₂) (union f m₂ m₁) union-comm-subset : ∀ (m₁ m₂ : Map) → subset (union f m₁ m₂) (union f m₂ m₁)
union-comm-subset m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂ union-comm-subset m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂
with Expr-Provenance f k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂) with Expr-Provenance f k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
... | (_ , (bothᵘ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂)) ... | (_ , (bothᵘ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂))
@ -374,63 +376,63 @@ module _ (f : B → B → B) where
(f v₂ v₁ , (f-comm v₁ v₂ , I.union-combines u₂ u₁ v₂∈m₂ v₁∈m₁)) (f v₂ v₁ , (f-comm v₁ v₂ , I.union-combines u₂ u₁ v₂∈m₂ v₁∈m₁))
... | (_ , (in₁ {v₁} (single v₁∈m₁) k∉km₂ , v₁∈m₁m₂)) ... | (_ , (in₁ {v₁} (single v₁∈m₁) k∉km₂ , v₁∈m₁m₂))
rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁∈m₁m₂ = rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁∈m₁m₂ =
(v₁ , (refl , I.union-preserves-∈₂ k∉km₂ v₁∈m₁)) (v₁ , (≈-refl , I.union-preserves-∈₂ k∉km₂ v₁∈m₁))
... | (_ , (in₂ {v₂} k∉km₁ (single v₂∈m₂) , v₂∈m₁m₂)) ... | (_ , (in₂ {v₂} k∉km₁ (single v₂∈m₂) , v₂∈m₁m₂))
rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₂∈m₁m₂ = rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₂∈m₁m₂ =
(v₂ , (refl , I.union-preserves-∈₁ u₂ v₂∈m₂ k∉km₁)) (v₂ , (≈-refl , I.union-preserves-∈₁ u₂ v₂∈m₂ k∉km₁))
module _ (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≡ f b₁ (f b₂ b₃)) where module _ (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≈ f b₁ (f b₂ b₃)) where
union-assoc : ∀ (m₁ m₂ m₃ : Map) → lift (_≡_) (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃)) union-assoc : ∀ (m₁ m₂ m₃ : Map) → lift (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃))
union-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (union-assoc₁ , union-assoc₂) union-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (union-assoc₁ , union-assoc₂)
where where
union-assoc₁ : subset (_≡_) (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃)) union-assoc₁ : subset (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃))
union-assoc₁ k v k,v∈m₁₂m₃ union-assoc₁ k v k,v∈m₁₂m₃
with Expr-Provenance f k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃) with Expr-Provenance f k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃)
... | (_ , (in₂ k∉ke₁₂ (single {v₃} v₃∈e₃) , v₃∈m₁₂m₃)) ... | (_ , (in₂ k∉ke₁₂ (single {v₃} v₃∈e₃) , v₃∈m₁₂m₃))
rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₃∈m₁₂m₃ = rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₃∈m₁₂m₃ =
let (k∉ke₁ , k∉ke₂) = I.∉-union-∉-either {l₁ = l₁} {l₂ = l₂} k∉ke₁₂ let (k∉ke₁ , k∉ke₂) = I.∉-union-∉-either {l₁ = l₁} {l₂ = l₂} k∉ke₁₂
in (v₃ , (refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃))) in (v₃ , (≈-refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃)))
... | (_ , (in₁ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) k∉ke₃ , v₂∈m₁₂m₃)) ... | (_ , (in₁ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) k∉ke₃ , v₂∈m₁₂m₃))
rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂∈m₁₂m₃ = rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂∈m₁₂m₃ =
(v₂ , (refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃))) (v₂ , (≈-refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃)))
... | (_ , (bothᵘ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₂v₃∈m₁₂m₃)) ... | (_ , (bothᵘ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₂v₃∈m₁₂m₃))
rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂v₃∈m₁₂m₃ = rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂v₃∈m₁₂m₃ =
(f v₂ v₃ , (refl , I.union-preserves-∈₂ k∉ke₁ (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃))) (f v₂ v₃ , (≈-refl , I.union-preserves-∈₂ k∉ke₁ (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
... | (_ , (in₁ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) k∉ke₃ , v₁∈m₁₂m₃)) ... | (_ , (in₁ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) k∉ke₃ , v₁∈m₁₂m₃))
rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁∈m₁₂m₃ = rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁∈m₁₂m₃ =
(v₁ , (refl , I.union-preserves-∈₁ u₁ v₁∈e₁ (I.union-preserves-∉ k∉ke₂ k∉ke₃))) (v₁ , (≈-refl , I.union-preserves-∈₁ u₁ v₁∈e₁ (I.union-preserves-∉ k∉ke₂ k∉ke₃)))
... | (_ , (bothᵘ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) (single {v₃} v₃∈e₃) , v₁v₃∈m₁₂m₃)) ... | (_ , (bothᵘ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) (single {v₃} v₃∈e₃) , v₁v₃∈m₁₂m₃))
rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₃∈m₁₂m₃ = rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₃∈m₁₂m₃ =
(f v₁ v₃ , (refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃))) (f v₁ v₃ , (≈-refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃)))
... | (_ , (in₁ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) k∉ke₃), v₁v₂∈m₁₂m₃) ... | (_ , (in₁ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) k∉ke₃), v₁v₂∈m₁₂m₃)
rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂∈m₁₂m₃ = rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂∈m₁₂m₃ =
(f v₁ v₂ , (refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃))) (f v₁ v₂ , (≈-refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃)))
... | (_ , (bothᵘ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃)) ... | (_ , (bothᵘ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃))
rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ = rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ =
(f v₁ (f v₂ v₃) , (f-assoc v₁ v₂ v₃ , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃))) (f v₁ (f v₂ v₃) , (f-assoc v₁ v₂ v₃ , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
union-assoc₂ : subset (_≡_) (union f m₁ (union f m₂ m₃)) (union f (union f m₁ m₂) m₃) union-assoc₂ : subset (union f m₁ (union f m₂ m₃)) (union f (union f m₁ m₂) m₃)
union-assoc₂ k v k,v∈m₁m₂₃ union-assoc₂ k v k,v∈m₁m₂₃
with Expr-Provenance f k ((` m₁) ∪ ((` m₂) ∪ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃) with Expr-Provenance f k ((` m₁) ∪ ((` m₂) ∪ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃)
... | (_ , (in₂ k∉ke₁ (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₃∈m₁m₂₃)) ... | (_ , (in₂ k∉ke₁ (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₃∈m₁m₂₃))
rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₃∈m₁m₂₃ = rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₃∈m₁m₂₃ =
(v₃ , (refl , I.union-preserves-∈₂ (I.union-preserves-∉ k∉ke₁ k∉ke₂) v₃∈e₃)) (v₃ , (≈-refl , I.union-preserves-∈₂ (I.union-preserves-∉ k∉ke₁ k∉ke₂) v₃∈e₃))
... | (_ , (in₂ k∉ke₁ (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₂∈m₁m₂₃)) ... | (_ , (in₂ k∉ke₁ (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₂∈m₁m₂₃))
rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂∈m₁m₂₃ = rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂∈m₁m₂₃ =
(v₂ , (refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) k∉ke₃)) (v₂ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) k∉ke₃))
... | (_ , (in₂ k∉ke₁ (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₂v₃∈m₁m₂₃)) ... | (_ , (in₂ k∉ke₁ (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₂v₃∈m₁m₂₃))
rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂v₃∈m₁m₂₃ = rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂v₃∈m₁m₂₃ =
(f v₂ v₃ , (refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) v₃∈e₃)) (f v₂ v₃ , (≈-refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) v₃∈e₃))
... | (_ , (in₁ (single {v₁} v₁∈e₁) k∉ke₂₃ , v₁∈m₁m₂₃)) ... | (_ , (in₁ (single {v₁} v₁∈e₁) k∉ke₂₃ , v₁∈m₁m₂₃))
rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁∈m₁m₂₃ = rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁∈m₁m₂₃ =
let (k∉ke₂ , k∉ke₃) = I.∉-union-∉-either {l₁ = l₂} {l₂ = l₃} k∉ke₂₃ let (k∉ke₂ , k∉ke₃) = I.∉-union-∉-either {l₁ = l₂} {l₂ = l₃} k∉ke₂₃
in (v₁ , (refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) k∉ke₃)) in (v₁ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) k∉ke₃))
... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₁v₃∈m₁m₂₃)) ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₁v₃∈m₁m₂₃))
rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₃∈m₁m₂₃ = rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₃∈m₁m₂₃ =
(f v₁ v₃ , (refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) v₃∈e₃)) (f v₁ v₃ , (≈-refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) v₃∈e₃))
... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₁v₂∈m₁m₂₃)) ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₁v₂∈m₁m₂₃))
rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂∈m₁m₂₃ = rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂∈m₁m₂₃ =
(f v₁ v₂ , (refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) k∉ke₃)) (f v₁ v₂ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) k∉ke₃))
... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃)) ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃))
rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ = rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ =
(f (f v₁ v₂) v₃ , (sym (f-assoc v₁ v₂ v₃) , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃)) (f (f v₁ v₂) v₃ , (≈-sym (f-assoc v₁ v₂ v₃) , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃))