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Use an expression-based provenance to make enumerating cases easier

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This should come in handy for the associativity proof.
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Danila Fedorin 2023-07-30 16:43:07 -07:00
parent 6039c1dfab
commit de2f202bdf
1 changed files with 33 additions and 29 deletions
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Map.agda
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@ -265,6 +265,10 @@ data Provenance (k : A) (m₁ m₂ : Map) : Set (a ⊔ b) where
in₁ : (v₁ : B) → (k , v₁) ∈ m₁ → ¬ k ∈k m₂ → Provenance k m₁ m₂
in₂ : (v₂ : B) → ¬ k ∈k m₁ → (k , v₂) ∈ m₂ → Provenance k m₁ m₂
data Expr : Set (a ⊔ b) where
`_ : Map → Expr
_∪_ : Expr → Expr → Expr
module _ (f : B → B → B) where
open ImplInsert f renaming
( insert to insert-impl
@ -277,31 +281,32 @@ module _ (f : B → B → B) where
union : Map → Map → Map
union (kvs₁ , _) (kvs₂ , uks₂) = (union-impl kvs₁ kvs₂ , union-preserves-Unique kvs₁ kvs₂ uks₂)
MergeResult : {k : A} {m₁ m₂ : Map} → Provenance k m₁ m₂ → Set (a ⊔ b)
MergeResult {k} {m₁} {m₂} (both v₁ v₂ _ _) = (k , f v₁ v₂) ∈ union m₁ m₂
MergeResult {k} {m₁} {m₂} (in₁ v₁ _ _) = (k , v₁) ∈ union m₁ m₂
MergeResult {k} {m₁} {m₂} (in₂ v₂ _ _) = (k , v₂) ∈ union m₁ m₂
⟦_⟧ : Expr -> Map
⟦ ` m ⟧ = m
⟦ e₁ ∪ e₂ ⟧ = union ⟦ e₁ ⟧ ⟦ e₂ ⟧
data Magic (k : A) : B → Expr → Set (a ⊔ b) where
single : ∀ {v : B} {m : Map} → (k , v) ∈ m → Magic k v (` m)
in₁ᵘ : ∀ {v : B} {e₁ e₂ : Expr} → Magic k v e₁ → ¬ k ∈k ⟦ e₂ ⟧ → Magic k v (e₁ ∪ e₂)
in₂ᵘ : ∀ {v : B} {e₁ e₂ : Expr} → ¬ k ∈k ⟦ e₁ ⟧ → Magic k v e₂ → Magic k v (e₁ ∪ e₂)
bothᵘ : ∀ {v₁ v₂ : B} {e₁ e₂ : Expr} → Magic k v₁ e₁ → Magic k v₂ e₂ → Magic k (f v₁ v₂) (e₁ ∪ e₂)
Expr-Magic : ∀ (k : A) (e : Expr) → k ∈k ⟦ e ⟧ → Σ B (λ v → (Magic k v e × (k , v) ∈ ⟦ e ⟧))
Expr-Magic k (` m) k∈km = let (v , k,v∈m) = locate k∈km in (v , (single k,v∈m , k,v∈m))
Expr-Magic k (e₁ ∪ e₂) k∈ke₁e₂
with ∈k-dec k (proj₁ ⟦ e₁ ⟧) | ∈k-dec k (proj₁ ⟦ e₂ ⟧)
... | yes k∈ke₁ | yes k∈ke₂ =
let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Magic k e₁ k∈ke₁
(v₂ , (g₂ , k,v₂∈e₂)) = Expr-Magic k e₂ k∈ke₂
in (f v₁ v₂ , (bothᵘ g₁ g₂ , union-combines (proj₂ ⟦ e₁ ⟧) (proj₂ ⟦ e₂ ⟧) k,v₁∈e₁ k,v₂∈e₂))
... | yes k∈ke₁ | no k∉ke₂ =
let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Magic k e₁ k∈ke₁
in (v₁ , (in₁ᵘ g₁ k∉ke₂ , union-preserves-∈₂ (proj₂ ⟦ e₁ ⟧) k,v₁∈e₁ k∉ke₂))
... | no k∉ke₁ | yes k∈ke₂ =
let (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Magic k e₂ k∈ke₂
in (v₂ , (in₂ᵘ k∉ke₁ g₂ , union-preserves-∈₁ k∉ke₁ k,v₂∈e₂))
... | no k∉ke₁ | no k∉ke₂ = absurd (union-preserves-∉ k∉ke₁ k∉ke₂ k∈ke₁e₂)
union-provenance : ∀ (m₁ m₂ : Map) (k : A) → k ∈k union m₁ m₂ → Σ (Provenance k m₁ m₂) MergeResult
union-provenance m₁@(l₁ , u₁) m₂@(l₂ , u₂) k k∈km₁m₂
with ∈k-dec k l₁ | ∈k-dec k l₂
... | yes k∈kl₁ | yes k∈kl₂ =
let
(v₁ , k,v₁∈l₁) = locate k∈kl₁
(v₂ , k,v₂∈l₂) = locate k∈kl₂
in
(both v₁ v₂ k,v₁∈l₁ k,v₂∈l₂ , union-combines u₁ u₂ k,v₁∈l₁ k,v₂∈l₂)
... | yes k∈kl₁ | no k∉kl₂ =
let
(v₁ , k,v₁∈l₁) = locate k∈kl₁
in
(in₁ v₁ k,v₁∈l₁ k∉kl₂ , union-preserves-∈₂ u₁ k,v₁∈l₁ k∉kl₂)
... | no k∉kl₁ | yes k∈kl₂ =
let
(v₂ , k,v₂∈l₂) = locate k∈kl₂
in
(in₂ v₂ k∉kl₁ k,v₂∈l₂ , union-preserves-∈₁ k∉kl₁ k,v₂∈l₂)
... | no k∉kl₁ | no k∉kl₂ = absurd (union-preserves-∉ k∉kl₁ k∉kl₂ k∈km₁m₂)
module _ (_≈_ : B → B → Set b) where
open ImplRelation _≈_ renaming (subset to subset-impl)
@ -319,14 +324,13 @@ module _ (f : B → B → B) where
where
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
with union-provenance f m₁ m₂ k (∈-cong proj₁ k,v∈m₁m₂)
... | (both v₁ v₂ v₁∈m₁ v₂∈m₂ , v₁v₂∈m₁m₂)
with Expr-Magic f k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
... | (_ , (bothᵘ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂))
rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ =
(f v₂ v₁ , (f-comm v₁ v₂ , ImplInsert.union-combines f u₂ u₁ v₂∈m₂ v₁∈m₁))
... | (in₁ v₁ v₁∈m₁ k∉km₂ , v₁∈m₁m₂)
... | (_ , (in₁ᵘ {v₁} (single v₁∈m₁) k∉km₂ , v₁∈m₁m₂))
rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁∈m₁m₂ =
(v₁ , (refl , ImplInsert.union-preserves-∈₁ f k∉km₂ v₁∈m₁))
... | (in₂ v₂ k∉km₁ v₂∈m₂ , v₂∈m₁m₂)
... | (_ , (in₂ᵘ {v₂} k∉km₁ (single v₂∈m₂) , v₂∈m₁m₂))
rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₂∈m₁m₂ =
(v₂ , (refl , ImplInsert.union-preserves-∈₂ f u₂ v₂∈m₂ k∉km₁))