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Use inferred variables for proofs where possible

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin 2023-07-30 13:19:00 -07:00
parent 4033a1b33d
commit b066db9829
1 changed files with 59 additions and 61 deletions
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Map.agda
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@ -119,129 +119,128 @@ private module ImplInsert (f : B → B → B) where
merge : List (A × B) → List (A × B) → List (A × B)
merge m₁ m₂ = foldr insert m₂ m₁
insert-keys-∈ : ∀ (k : A) (v : B) (l : List (A × B)) →
insert-keys-∈ : ∀ {k : A} {v : B} {l : List (A × B)} →
k ∈k l → keys l ≡ keys (insert k v l)
insert-keys-∈ k v ((k' , v') ∷ xs) (here k≡k')
insert-keys-∈ {k} {v} {(k' , v') ∷ xs} (here k≡k')
with (≡-dec-A k k')
... | yes _ = refl
... | no k≢k' = absurd (k≢k' k≡k')
insert-keys-∈ k v ((k' , _) ∷ xs) (there k∈kxs)
insert-keys-∈ {k} {v} {(k' , _) ∷ xs} (there k∈kxs)
with (≡-dec-A k k')
... | yes _ = refl
... | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∈ k v xs k∈kxs)
... | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∈ k∈kxs)
insert-keys-∉ : ∀ (k : A) (v : B) (l : List (A × B)) →
insert-keys-∉ : ∀ {k : A} {v : B} {l : List (A × B)} →
¬ (k ∈k l) → (keys l ++ (k ∷ [])) ≡ keys (insert k v l)
insert-keys-∉ k v [] _ = refl
insert-keys-∉ k v ((k' , v') ∷ xs) k∉kl
insert-keys-∉ {k} {v} {[]} _ = refl
insert-keys-∉ {k} {v} {(k' , v') ∷ xs} k∉kl
with (≡-dec-A k k')
... | yes k≡k' = absurd (k∉kl (here k≡k'))
... | no _ = cong (λ xs' → k' ∷ xs')
(insert-keys-∉ k v xs (λ k∈kxs → k∉kl (there k∈kxs)))
(insert-keys-∉ (λ k∈kxs → k∉kl (there k∈kxs)))
insert-preserves-Unique : ∀ (k : A) (v : B) (l : List (A × B))
insert-preserves-Unique : ∀ {k : A} {v : B} {l : List (A × B)}
→ Unique (keys l) → Unique (keys (insert k v l))
insert-preserves-Unique k v l u
insert-preserves-Unique {k} {v} {l} u
with (∈k-dec k l)
... | yes k∈kl rewrite insert-keys-∈ k v l k∈kl = u
... | no k∉kl rewrite sym (insert-keys-∉ k v l k∉kl) = Unique-append k∉kl u
... | yes k∈kl rewrite insert-keys-∈ {v = v} k∈kl = u
... | no k∉kl rewrite sym (insert-keys-∉ {v = v} k∉kl) = Unique-append k∉kl u
merge-preserves-Unique : ∀ (l₁ l₂ : List (A × B)) →
Unique (keys l₂) → Unique (keys (merge l₁ l₂))
merge-preserves-Unique [] l₂ u₂ = u₂
merge-preserves-Unique ((k₁ , v₁) ∷ xs₁) l₂ u₂ =
insert-preserves-Unique k₁ v₁ (merge xs₁ l₂)
(merge-preserves-Unique xs₁ l₂ u₂)
insert-preserves-Unique (merge-preserves-Unique xs₁ l₂ u₂)
insert-preserves-∈-right : ∀ (k k' : A) (v v' : B) (l : List (A × B)) →
insert-preserves-∈-right : ∀ {k k' : A} {v v' : B} {l : List (A × B)} →
¬ k ≡ k' → (k , v) ∈ l → (k , v) ∈ insert k' v' l
insert-preserves-∈-right k k' v v' (x ∷ xs) k≢k' (here k,v=x)
insert-preserves-∈-right {k} {k'} {l = x ∷ xs} k≢k' (here k,v=x)
rewrite sym k,v=x with ≡-dec-A k' k
... | yes k'≡k = absurd (k≢k' (sym k'≡k))
... | no _ = here refl
insert-preserves-∈-right k k' v v' ((k'' , _) ∷ xs) k≢k' (there k,v∈xs)
insert-preserves-∈-right {k} {k'} {l = (k'' , _) ∷ xs} k≢k' (there k,v∈xs)
with ≡-dec-A k' k''
... | yes _ = there k,v∈xs
... | no _ = there (insert-preserves-∈-right k k' v v' xs k≢k' k,v∈xs)
... | no _ = there (insert-preserves-∈-right k≢k' k,v∈xs)
insert-preserves-∈k-right : ∀ (k k' : A) (v' : B) (l : List (A × B)) →
insert-preserves-∈k-right : ∀ {k k' : A} {v' : B} {l : List (A × B)} →
¬ k ≡ k' → k ∈k l → k ∈k insert k' v' l
insert-preserves-∈k-right k k' v' l k≢k' k∈kl =
insert-preserves-∈k-right k≢k' k∈kl =
let (v , k,v∈l) = locate k∈kl
in ∈-cong proj₁ (insert-preserves-∈-right k k' v v' l k≢k' k,v∈l)
in ∈-cong proj₁ (insert-preserves-∈-right k≢k' k,v∈l)
insert-preserves-∉-right : ∀ (k k' : A) (v' : B) (l : List (A × B)) →
insert-preserves-∉-right : ∀ {k k' : A} {v' : B} {l : List (A × B)} →
¬ k ≡ k' → ¬ k ∈k l → ¬ k ∈k insert k' v' l
insert-preserves-∉-right k k' v' [] k≢k' k∉kl (here k≡k') = k≢k' k≡k'
insert-preserves-∉-right k k' v' [] k≢k' k∉kl (there ())
insert-preserves-∉-right k k' v' ((k'' , v'') ∷ xs) k≢k' k∉kl k∈kil
insert-preserves-∉-right {l = []} k≢k' k∉kl (here k≡k') = k≢k' k≡k'
insert-preserves-∉-right {l = []} k≢k' k∉kl (there ())
insert-preserves-∉-right {k} {k'} {v'} {(k'' , v'') ∷ xs} k≢k' k∉kl k∈kil
with ≡-dec-A k k''
... | yes k≡k'' = k∉kl (here k≡k'')
... | no k≢k'' with ≡-dec-A k' k'' | k∈kil
... | yes k'≡k'' | here k≡k'' = k≢k'' k≡k''
... | yes k'≡k'' | there k∈kxs = k∉kl (there k∈kxs)
... | no k'≢k'' | here k≡k'' = k∉kl (here k≡k'')
... | no k'≢k'' | there k∈kxs = insert-preserves-∉-right k k' v' xs k≢k'
... | no k'≢k'' | there k∈kxs = insert-preserves-∉-right k≢k'
(λ k∈kxs → k∉kl (there k∈kxs)) k∈kxs
merge-preserves-∉ : ∀ (k : A) (l₁ l₂ : List (A × B)) →
merge-preserves-∉ : ∀ {k : A} {l₁ l₂ : List (A × B)} →
¬ k ∈k l₁ → ¬ k ∈k l₂ → ¬ k ∈k merge l₁ l₂
merge-preserves-∉ k [] l₂ _ k∉kl₂ = k∉kl₂
merge-preserves-∉ k ((k' , v') ∷ xs₁) l₂ k∉kl₁ k∉kl₂
merge-preserves-∉ {l₁ = []} _ k∉kl₂ = k∉kl₂
merge-preserves-∉ {k} {(k' , v') ∷ xs₁} k∉kl₁ k∉kl₂
with ≡-dec-A k k'
... | yes k≡k' = absurd (k∉kl₁ (here k≡k'))
... | no k≢k' = insert-preserves-∉-right k k' v' _ k≢k' (merge-preserves-∉ k xs₁ l₂ (λ k∈kxs₁ → k∉kl₁ (there k∈kxs₁)) k∉kl₂)
... | no k≢k' = insert-preserves-∉-right k≢k' (merge-preserves-∉ (λ k∈kxs₁ → k∉kl₁ (there k∈kxs₁)) k∉kl₂)
merge-preserves-keys₁ : ∀ (k : A) (v : B) (l₁ l₂ : List (A × B)) →
merge-preserves-keys₁ : ∀ {k : A} {v : B} {l₁ l₂ : List (A × B)} →
¬ k ∈k l₁ → (k , v) ∈ l₂ → (k , v) ∈ merge l₁ l₂
merge-preserves-keys₁ k v [] l₂ _ k,v∈l₂ = k,v∈l₂
merge-preserves-keys₁ k v ((k' , v') ∷ xs₁) l₂ k∉kl₁ k,v∈l₂ =
let recursion = merge-preserves-keys₁ k v xs₁ l₂ (λ k∈xs₁ → k∉kl₁ (there k∈xs₁)) k,v∈l₂
in insert-preserves-∈-right k k' v v' _ (λ k≡k' → k∉kl₁ (here k≡k')) recursion
merge-preserves-keys₁ {l₁ = []} _ k,v∈l₂ = k,v∈l₂
merge-preserves-keys₁ {l₁ = (k' , v') ∷ xs₁} k∉kl₁ k,v∈l₂ =
let recursion = merge-preserves-keys₁ (λ k∈xs₁ → k∉kl₁ (there k∈xs₁)) k,v∈l₂
in insert-preserves-∈-right (λ k≡k' → k∉kl₁ (here k≡k')) recursion
insert-fresh : ∀ (k : A) (v : B) (l : List (A × B)) →
insert-fresh : ∀ {k : A} {v : B} {l : List (A × B)} →
¬ k ∈k l → (k , v) ∈ insert k v l
insert-fresh k v [] k∉kl = here refl
insert-fresh k v ((k' , v') ∷ xs) k∉kl
insert-fresh {l = []} k∉kl = here refl
insert-fresh {k} {l = (k' , v') ∷ xs} k∉kl
with ≡-dec-A k k'
... | yes k≡k' = absurd (k∉kl (here k≡k'))
... | no _ = there (insert-fresh k v xs (λ k∈kxs → k∉kl (there k∈kxs)))
... | no _ = there (insert-fresh (λ k∈kxs → k∉kl (there k∈kxs)))
merge-preserves-keys₂ : ∀ (k : A) (v : B) (l₁ l₂ : List (A × B)) →
merge-preserves-keys₂ : ∀ {k : A} {v : B} {l₁ l₂ : List (A × B)} →
Unique (keys l₁) → (k , v) ∈ l₁ → ¬ k ∈k l₂ → (k , v) ∈ merge l₁ l₂
merge-preserves-keys₂ k v ((k' , v') ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) (there k,v∈xs₁) k∉kl₂ =
insert-preserves-∈-right k k' v v' (merge xs₁ l₂) k≢k' k,v∈mxs₁l
merge-preserves-keys₂ {k} {v} {(k' , v') ∷ xs₁} (push k'≢xs₁ uxs₁) (there k,v∈xs₁) k∉kl₂ =
insert-preserves-∈-right k≢k' k,v∈mxs₁l
where
k,v∈mxs₁l = merge-preserves-keys₂ k v xs₁ l₂ uxs₁ k,v∈xs₁ k∉kl₂
k,v∈mxs₁l = merge-preserves-keys₂ uxs₁ k,v∈xs₁ k∉kl₂
k≢k' : ¬ k ≡ k'
k≢k' with ≡-dec-A k k'
... | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs₁ (∈-cong proj₁ k,v∈xs₁))
... | no k≢k' = k≢k'
merge-preserves-keys₂ k v ((k' , v') ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) (here k,v≡k',v') k∉kl₂
merge-preserves-keys₂ {l₁ = (k' , v') ∷ xs₁} (push k'≢xs₁ uxs₁) (here k,v≡k',v') k∉kl₂
rewrite cong proj₁ k,v≡k',v' rewrite cong proj₂ k,v≡k',v' =
insert-fresh k' v' _ (merge-preserves-∉ k' xs₁ l₂ (All¬-¬Any k'≢xs₁) k∉kl₂)
insert-fresh (merge-preserves-∉ (All¬-¬Any k'≢xs₁) k∉kl₂)
insert-combines : ∀ (k : A) (v v' : B) (l : List (A × B)) →
insert-combines : ∀ {k : A} {v v' : B} {l : List (A × B)} →
Unique (keys l) → (k , v') ∈ l → (k , f v v') ∈ (insert k v l)
insert-combines k v v' ((k' , v'') ∷ xs) _ (here k,v'≡k',v'')
insert-combines {l = (k' , v'') ∷ xs} _ (here k,v'≡k',v'')
rewrite cong proj₁ k,v'≡k',v'' rewrite cong proj₂ k,v'≡k',v''
with ≡-dec-A k' k'
... | yes _ = here refl
... | no k≢k' = absurd (k≢k' refl)
insert-combines k v v' ((k' , v'') ∷ xs) (push k'≢xs uxs) (there k,v'∈xs)
insert-combines {k} {l = (k' , v'') ∷ xs} (push k'≢xs uxs) (there k,v'∈xs)
with ≡-dec-A k k'
... | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs (∈-cong proj₁ k,v'∈xs))
... | no k≢k' = there (insert-combines k v v' xs uxs k,v'∈xs)
... | no k≢k' = there (insert-combines uxs k,v'∈xs)
merge-combines : forall (k : A) (v₁ v₂ : B) (l₁ l₂ : List (A × B)) →
merge-combines : forall {k : A} {v₁ v₂ : B} {l₁ l₂ : List (A × B)} →
Unique (keys l₁) → Unique (keys l₂) →
(k , v₁) ∈ l₁ → (k , v₂) ∈ l₂ → (k , f v₁ v₂) ∈ merge l₁ l₂
merge-combines k v₁ v₂ ((k' , v) ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) ul₂ (here k,v₁≡k',v) k,v₂∈l₂
merge-combines {l₁ = (k' , v) ∷ xs₁} {l₂} (push k'≢xs₁ uxs₁) ul₂ (here k,v₁≡k',v) k,v₂∈l₂
rewrite cong proj₁ (sym (k,v₁≡k',v)) rewrite cong proj₂ (sym (k,v₁≡k',v)) =
insert-combines k v₁ v₂ _ (merge-preserves-Unique xs₁ l₂ ul₂) (merge-preserves-keys₁ k v₂ xs₁ l₂ (All¬-¬Any k'≢xs₁) k,v₂∈l₂)
merge-combines k v₁ v₂ ((k' , v) ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) ul₂ (there k,v₁∈xs₁) k,v₂∈l₂ =
insert-preserves-∈-right k k' (f v₁ v₂) v _ k≢k' (merge-combines k v₁ v₂ xs₁ l₂ uxs₁ ul₂ k,v₁∈xs₁ k,v₂∈l₂)
insert-combines (merge-preserves-Unique xs₁ l₂ ul₂) (merge-preserves-keys₁ (All¬-¬Any k'≢xs₁) k,v₂∈l₂)
merge-combines {k} {l₁ = (k' , v) ∷ xs₁} (push k'≢xs₁ uxs₁) ul₂ (there k,v₁∈xs₁) k,v₂∈l₂ =
insert-preserves-∈-right k≢k' (merge-combines uxs₁ ul₂ k,v₁∈xs₁ k,v₂∈l₂)
where
k≢k' : ¬ k ≡ k'
k≢k' with ≡-dec-A k k'
@ -273,7 +272,7 @@ module _ (f : B → B → B) where
)
insert : A → B → Map → Map
insert k v (kvs , uks) = (insert-impl k v kvs , insert-preserves-Unique k v kvs uks)
insert k v (kvs , uks) = (insert-impl k v kvs , insert-preserves-Unique uks)
merge : Map → Map → Map
merge (kvs₁ , _) (kvs₂ , uks₂) = (merge-impl kvs₁ kvs₂ , merge-preserves-Unique kvs₁ kvs₂ uks₂)
@ -291,18 +290,18 @@ module _ (f : B → B → B) where
(v₁ , k,v₁∈l₁) = locate k∈kl₁
(v₂ , k,v₂∈l₂) = locate k∈kl₂
in
(both v₁ v₂ k,v₁∈l₁ k,v₂∈l₂ , merge-combines k v₁ v₂ l₁ l₂ u₁ u₂ k,v₁∈l₁ k,v₂∈l₂)
(both v₁ v₂ k,v₁∈l₁ k,v₂∈l₂ , merge-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 , merge-preserves-keys₂ k v₁ l₁ l₂ u₁ k,v₁∈l₁ k∉kl₂)
(in₁ v₁ k,v₁∈l₁ k∉kl₂ , merge-preserves-keys₂ 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 , merge-preserves-keys₁ k v₂ l₁ l₂ k∉kl₁ k,v₂∈l₂)
... | no k∉kl₁ | no k∉kl₂ = absurd (merge-preserves-∉ k l₁ l₂ k∉kl₁ k∉kl₂ k∈km₁m₂)
(in₂ v₂ k∉kl₁ k,v₂∈l₂ , merge-preserves-keys₁ k∉kl₁ k,v₂∈l₂)
... | no k∉kl₁ | no k∉kl₂ = absurd (merge-preserves-∉ k∉kl₁ k∉kl₂ k∈km₁m₂)
module _ (_≈_ : B → B → Set b) where
open ImplRelation _≈_ renaming (subset to subset-impl)
@ -319,8 +318,7 @@ module _ (f : B → B → B) where
merge-comm m₁ m₂ = (merge-comm-subset m₁ m₂ , merge-comm-subset m₂ m₁)
where
merge-comm-subset : ∀ (m₁ m₂ : Map) → subset (_≡_) (merge f m₁ m₂) (merge f m₂ m₁)
merge-comm-subset m₁ m₂ k v k,v∈m₁m₂
with ∈k-dec k (proj₁ (merge f m₂ m₁) )
merge-comm-subset m₁ m₂ k v k,v∈m₁m₂ with ∈k-dec k (proj₁ (merge f m₂ m₁) )
... | no k∉km₂m₁ = {!!}
... | yes k∈km₂m₁ with merge-provenance f m₁ m₂ k (∈-cong proj₁ k,v∈m₁m₂) | merge-provenance f m₂ m₁ k k∈km₂m₁
... | (both v₁ v₂ v₁∈m₁ v₂∈m₂ , v₁v₂∈m₁m₂) | (both v₂' v₁' v₂'∈m₂ v₁'∈m₁ , v₂'v₁'∈m₂m₁)