More cleanup to FiniteValueMap
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -93,17 +93,20 @@ module IterProdIsomorphism where
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_⊆ᵐ_ : ∀ {ks₁ ks₂ : List A} → FiniteMap ks₁ → FiniteMap ks₂ → Set
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_⊆ᵐ_ fm₁ fm₂ = subset-impl (proj₁ (proj₁ fm₁)) (proj₁ (proj₁ fm₂))
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_≈ⁱᵖ_ : ∀ {ks : List A} → IterProd (length ks) → IterProd (length ks) → Set
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_≈ⁱᵖ_ {ks} = IP._≈_ (length ks)
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_≈ⁱᵖ_ : ∀ {n : ℕ} → IterProd n → IterProd n → Set
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_≈ⁱᵖ_ {n} = IP._≈_ n
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_⊔ⁱᵖ_ : ∀ {ks : List A} → IterProd (length ks) → IterProd (length ks) → IterProd (length ks)
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_⊔ⁱᵖ_ : ∀ {ks : List A} →
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IterProd (length ks) → IterProd (length ks) → IterProd (length ks)
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_⊔ⁱᵖ_ {ks} = IP._⊔_ (length ks)
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_∈ᵐ_ : ∀ {ks : List A} → A × B → FiniteMap ks → Set
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_∈ᵐ_ {ks} k,v fm = k,v ∈ proj₁ fm
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-- The left inverse is: from (to x) = x
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from-to-inverseˡ : ∀ {ks : List A} (uks : Unique ks) →
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Inverseˡ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {ks}) (from {ks}) (to {ks} uks) -- from (to x) = x
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Inverseˡ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {length ks})
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(from {ks}) (to {ks} uks)
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from-to-inverseˡ {[]} _ _ = IsEquivalence.≈-refl (IP.≈-equiv 0)
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from-to-inverseˡ {k ∷ ks'} (push k≢ks' uks') (v , rest)
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with ((fm' , ufm') , refl) ← to uks' rest in p rewrite sym p =
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@ -115,33 +118,47 @@ module IterProdIsomorphism where
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-- The map has its own uniqueness proof, but the call to 'to' needs a standalone
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-- uniqueness proof too. Work with both proofs as needed to thread things through.
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--
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-- The right inverse is: to (from x) = x
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from-to-inverseʳ : ∀ {ks : List A} (uks : Unique ks) →
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Inverseʳ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {ks}) (from {ks}) (to {ks} uks) -- to (from x) = x
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from-to-inverseʳ {[]} _ (([] , empty) , kvs≡ks) rewrite kvs≡ks = ((λ k v ()) , (λ k v ()))
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from-to-inverseʳ {k ∷ ks'} uks@(push k≢ks'₁ uks'₁) fm₁@(m₁@((k , v) ∷ fm'₁ , push k≢ks'₂ uks'₂) , refl)
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with to uks'₁ (from ((fm'₁ , uks'₂) , refl)) | from-to-inverseʳ {ks'} uks'₁ ((fm'₁ , uks'₂) , refl)
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... | ((fm'₂ , ufm'₂) , _) | (fm'₂⊆fm'₁ , fm'₁⊆fm'₂) = (m₂⊆m₁ , m₁⊆m₂)
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Inverseʳ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {length ks})
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(from {ks}) (to {ks} uks)
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from-to-inverseʳ {[]} _ (([] , empty) , kvs≡ks) rewrite kvs≡ks =
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( (λ k v ())
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, (λ k v ())
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)
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from-to-inverseʳ {k ∷ ks'} uks@(push _ uks'₁) fm₁@(((k , v) ∷ fm'₁ , push _ uks'₂) , refl)
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with to uks'₁ (from ((fm'₁ , uks'₂) , refl))
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| from-to-inverseʳ {ks'} uks'₁ ((fm'₁ , uks'₂) , refl)
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... | ((fm'₂ , ufm'₂) , _)
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| (fm'₂⊆fm'₁ , fm'₁⊆fm'₂) = (m₂⊆m₁ , m₁⊆m₂)
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where
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kvs₁ = (k , v) ∷ fm'₁
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kvs₂ = (k , v) ∷ fm'₂
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m₁⊆m₂ : subset-impl kvs₁ kvs₂
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m₁⊆m₂ k' v' (here refl) = (v' , (IsLattice.≈-refl lB , here refl))
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m₁⊆m₂ k' v' (here refl) =
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(v' , (IsLattice.≈-refl lB , here refl))
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m₁⊆m₂ k' v' (there k',v'∈fm'₁) =
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let (v'' , (v'≈v'' , k',v''∈fm'₂)) = fm'₁⊆fm'₂ k' v' k',v'∈fm'₁
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let (v'' , (v'≈v'' , k',v''∈fm'₂)) =
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fm'₁⊆fm'₂ k' v' k',v'∈fm'₁
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in (v'' , (v'≈v'' , there k',v''∈fm'₂))
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m₂⊆m₁ : subset-impl kvs₂ kvs₁
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m₂⊆m₁ k' v' (here refl) = (v' , (IsLattice.≈-refl lB , here refl))
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m₂⊆m₁ k' v' (here refl) =
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(v' , (IsLattice.≈-refl lB , here refl))
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m₂⊆m₁ k' v' (there k',v'∈fm'₂) =
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let (v'' , (v'≈v'' , k',v''∈fm'₁)) = fm'₂⊆fm'₁ k' v' k',v'∈fm'₂
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let (v'' , (v'≈v'' , k',v''∈fm'₁)) =
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fm'₂⊆fm'₁ k' v' k',v'∈fm'₂
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in (v'' , (v'≈v'' , there k',v''∈fm'₁))
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private
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first-key-in-map : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → Σ B (λ v → (k , v) ∈ proj₁ fm)
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first-key-in-map : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) →
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Σ B (λ v → (k , v) ∈ proj₁ fm)
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first-key-in-map (((k , v) ∷ _ , _) , refl) = (v , here refl)
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from-first-value : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → proj₁ (from fm) ≡ proj₁ (first-key-in-map fm)
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from-first-value : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) →
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proj₁ (from fm) ≡ proj₁ (first-key-in-map fm)
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from-first-value {k} {ks} (((k , v) ∷ _ , push _ _) , refl) = refl
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-- We need pop because reasoning about two distinct 'refl' pattern
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@ -151,91 +168,151 @@ module IterProdIsomorphism where
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pop : ∀ {k : A} {ks : List A} → FiniteMap (k ∷ ks) → FiniteMap ks
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pop (((_ ∷ fm') , push _ ufm') , refl) = ((fm' , ufm') , refl)
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pop-≈ : ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) → fm₁ ≈ᵐ fm₂ → pop fm₁ ≈ᵐ pop fm₂
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pop-≈ {k} {ks} fm₁ fm₂ (fm₁⊆fm₂ , fm₂⊆fm₁) = (narrow fm₁⊆fm₂ , narrow fm₂⊆fm₁)
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pop-≈ : ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) →
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fm₁ ≈ᵐ fm₂ → pop fm₁ ≈ᵐ pop fm₂
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pop-≈ {k} {ks} fm₁ fm₂ (fm₁⊆fm₂ , fm₂⊆fm₁) =
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(narrow fm₁⊆fm₂ , narrow fm₂⊆fm₁)
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where
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narrow₁ : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} → fm₁ ⊆ᵐ fm₂ → pop fm₁ ⊆ᵐ fm₂
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narrow₁ {(_ ∷ _ , push _ _) , refl} kvs₁⊆kvs₂ k' v' k',v'∈fm'₁ = kvs₁⊆kvs₂ k' v' (there k',v'∈fm'₁)
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narrow₁ : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} →
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fm₁ ⊆ᵐ fm₂ → pop fm₁ ⊆ᵐ fm₂
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narrow₁ {(_ ∷ _ , push _ _) , refl} kvs₁⊆kvs₂ k' v' k',v'∈fm'₁ =
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kvs₁⊆kvs₂ k' v' (there k',v'∈fm'₁)
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narrow₂ : ∀ {fm₁ : FiniteMap ks} {fm₂ : FiniteMap (k ∷ ks)} → fm₁ ⊆ᵐ fm₂ → fm₁ ⊆ᵐ pop fm₂
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narrow₂ : ∀ {fm₁ : FiniteMap ks} {fm₂ : FiniteMap (k ∷ ks)} →
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fm₁ ⊆ᵐ fm₂ → fm₁ ⊆ᵐ pop fm₂
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narrow₂ {fm₁} {fm₂ = (_ ∷ fm'₂ , push k≢ks _) , kvs≡ks@refl} kvs₁⊆kvs₂ k' v' k',v'∈fm'₁
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with kvs₁⊆kvs₂ k' v' k',v'∈fm'₁
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... | (v'' , (v'≈v'' , here refl)) rewrite sym (proj₂ fm₁) = ⊥-elim (All¬-¬Any k≢ks (forget {m = proj₁ fm₁} k',v'∈fm'₁))
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... | (v'' , (v'≈v'' , there k',v'∈fm'₂)) = (v'' , (v'≈v'' , k',v'∈fm'₂))
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... | (v'' , (v'≈v'' , here refl)) rewrite sym (proj₂ fm₁) =
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⊥-elim (All¬-¬Any k≢ks (forget {m = proj₁ fm₁} k',v'∈fm'₁))
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... | (v'' , (v'≈v'' , there k',v'∈fm'₂)) =
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(v'' , (v'≈v'' , k',v'∈fm'₂))
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narrow : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} → fm₁ ⊆ᵐ fm₂ → pop fm₁ ⊆ᵐ pop fm₂
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narrow : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} →
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fm₁ ⊆ᵐ fm₂ → pop fm₁ ⊆ᵐ pop fm₂
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narrow {fm₁} {fm₂} x = narrow₂ {pop fm₁} (narrow₁ {fm₂ = fm₂} x)
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k,v∈pop⇒k,v∈ : ∀ {k : A} {ks : List A} {k' : A} {v : B} (fm : FiniteMap (k ∷ ks)) → (k' , v) ∈ᵐ pop fm → (¬ k ≡ k' × ((k' , v) ∈ᵐ fm))
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k,v∈pop⇒k,v∈ : ∀ {k : A} {ks : List A} {k' : A} {v : B} (fm : FiniteMap (k ∷ ks)) →
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(k' , v) ∈ᵐ pop fm → (¬ k ≡ k' × ((k' , v) ∈ᵐ fm))
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k,v∈pop⇒k,v∈ {k} {ks} {k'} {v} (m@((k , _) ∷ fm' , push k≢ks uks') , refl) k',v∈fm =
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((λ { refl → All¬-¬Any k≢ks (forget {m = (fm' , uks')} k',v∈fm) }), there k',v∈fm)
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( (λ { refl → All¬-¬Any k≢ks (forget {m = (fm' , uks')} k',v∈fm) })
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, there k',v∈fm
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)
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k,v∈⇒k,v∈pop : ∀ {k : A} {ks : List A} {k' : A} {v : B} (fm : FiniteMap (k ∷ ks)) → ¬ k ≡ k' → (k' , v) ∈ᵐ fm → (k' , v) ∈ᵐ pop fm
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k,v∈⇒k,v∈pop {k} {ks} {k'} {v} (m@((k , _) ∷ fm' , push k≢ks uks') , refl) k≢k' (here refl) = ⊥-elim (k≢k' refl)
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k,v∈⇒k,v∈pop {k} {ks} {k'} {v} (m@((k , _) ∷ fm' , push k≢ks uks') , refl) k≢k' (there k,v'∈fm') = k,v'∈fm'
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k,v∈⇒k,v∈pop : ∀ {k : A} {ks : List A} {k' : A} {v : B} (fm : FiniteMap (k ∷ ks)) →
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¬ k ≡ k' → (k' , v) ∈ᵐ fm → (k' , v) ∈ᵐ pop fm
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k,v∈⇒k,v∈pop (m@(_ ∷ _ , push k≢ks _) , refl) k≢k' (here refl) = ⊥-elim (k≢k' refl)
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k,v∈⇒k,v∈pop (m@(_ ∷ _ , push k≢ks _) , refl) k≢k' (there k,v'∈fm') = k,v'∈fm'
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Provenance-union : ∀ {ks : List A} (fm₁ fm₂ : FiniteMap ks) {k : A} {v : B} → (k , v) ∈ᵐ (fm₁ ⊔ᵐ fm₂) → Σ (B × B) (λ (v₁ , v₂) → ((v ≡ v₁ ⊔₂ v₂) × ((k , v₁) ∈ᵐ fm₁ × (k , v₂) ∈ᵐ fm₂)))
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FromBothMaps : ∀ (k : A) (v : B) {ks : List A} (fm₁ fm₂ : FiniteMap ks) → Set
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FromBothMaps k v fm₁ fm₂ =
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Σ (B × B)
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(λ (v₁ , v₂) → ( (v ≡ v₁ ⊔₂ v₂) × ((k , v₁) ∈ᵐ fm₁ × (k , v₂) ∈ᵐ fm₂)))
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Provenance-union : ∀ {ks : List A} (fm₁ fm₂ : FiniteMap ks) {k : A} {v : B} →
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(k , v) ∈ᵐ (fm₁ ⊔ᵐ fm₂) → FromBothMaps k v fm₁ fm₂
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Provenance-union fm₁@(m₁ , ks₁≡ks) fm₂@(m₂ , ks₂≡ks) {k} {v} k,v∈fm₁fm₂
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with Expr-Provenance k ((` m₁) ∪ (` m₂)) (forget {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂)
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... | (_ , (in₁ (single k,v∈m₁) k∉km₂ , _)) with k∈km₁ ← (forget {m = m₁} k,v∈m₁) rewrite trans ks₁≡ks (sym ks₂≡ks) = ⊥-elim (k∉km₂ k∈km₁)
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... | (_ , (in₂ k∉km₁ (single k,v∈m₂) , _)) with k∈km₂ ← (forget {m = m₂} k,v∈m₂) rewrite trans ks₁≡ks (sym ks₂≡ks) = ⊥-elim (k∉km₁ k∈km₂)
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... | (_ , (in₁ (single k,v∈m₁) k∉km₂ , _))
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with k∈km₁ ← (forget {m = m₁} k,v∈m₁)
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rewrite trans ks₁≡ks (sym ks₂≡ks) =
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⊥-elim (k∉km₂ k∈km₁)
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... | (_ , (in₂ k∉km₁ (single k,v∈m₂) , _))
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with k∈km₂ ← (forget {m = m₂} k,v∈m₂)
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rewrite trans ks₁≡ks (sym ks₂≡ks) =
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⊥-elim (k∉km₁ k∈km₂)
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... | (v₁⊔v₂ , (bothᵘ {v₁} {v₂} (single k,v₁∈m₁) (single k,v₂∈m₂) , k,v₁⊔v₂∈m₁m₂))
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rewrite Map-functional {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂ k,v₁⊔v₂∈m₁m₂ = ((v₁ , v₂) , (refl , (k,v₁∈m₁ , k,v₂∈m₂)))
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rewrite Map-functional {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂ k,v₁⊔v₂∈m₁m₂ =
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((v₁ , v₂) , (refl , (k,v₁∈m₁ , k,v₂∈m₂)))
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pop-⊔-distr : ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) → pop (fm₁ ⊔ᵐ fm₂) ≈ᵐ (pop fm₁ ⊔ᵐ pop fm₂)
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pop-⊔-distr {k} {ks} fm₁@(m₁ , _) fm₂@(m₂ , _) = (pfm₁fm₂⊆pfm₁pfm₂ , pfm₁pfm₂⊆pfm₁fm₂)
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pop-⊔-distr : ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) →
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pop (fm₁ ⊔ᵐ fm₂) ≈ᵐ (pop fm₁ ⊔ᵐ pop fm₂)
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pop-⊔-distr {k} {ks} fm₁@(m₁ , _) fm₂@(m₂ , _) =
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(pfm₁fm₂⊆pfm₁pfm₂ , pfm₁pfm₂⊆pfm₁fm₂)
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where
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-- pfm₁fm₂⊆pfm₁pfm₂ = {!!}
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pfm₁fm₂⊆pfm₁pfm₂ : pop (fm₁ ⊔ᵐ fm₂) ⊆ᵐ (pop fm₁ ⊔ᵐ pop fm₂)
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pfm₁fm₂⊆pfm₁pfm₂ k' v' k',v'∈pfm₁fm₂
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with (k≢k' , k',v'∈fm₁fm₂) ← k,v∈pop⇒k,v∈ (fm₁ ⊔ᵐ fm₂) k',v'∈pfm₁fm₂
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with ((v₁ , v₂) , (refl , (k,v₁∈fm₁ , k,v₂∈fm₂))) ← Provenance-union fm₁ fm₂ k',v'∈fm₁fm₂
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with ((v₁ , v₂) , (refl , (k,v₁∈fm₁ , k,v₂∈fm₂)))
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← Provenance-union fm₁ fm₂ k',v'∈fm₁fm₂
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with k',v₁∈pfm₁ ← k,v∈⇒k,v∈pop fm₁ k≢k' k,v₁∈fm₁
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with k',v₂∈pfm₂ ← k,v∈⇒k,v∈pop fm₂ k≢k' k,v₂∈fm₂
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= (v₁ ⊔₂ v₂ , (IsLattice.≈-refl lB , ⊔-combines {m₁ = proj₁ (pop fm₁)} {m₂ = proj₁ (pop fm₂)} k',v₁∈pfm₁ k',v₂∈pfm₂))
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=
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( v₁ ⊔₂ v₂
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, (IsLattice.≈-refl lB
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, ⊔-combines {m₁ = proj₁ (pop fm₁)}
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{m₂ = proj₁ (pop fm₂)}
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k',v₁∈pfm₁ k',v₂∈pfm₂
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)
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)
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pfm₁pfm₂⊆pfm₁fm₂ : (pop fm₁ ⊔ᵐ pop fm₂) ⊆ᵐ pop (fm₁ ⊔ᵐ fm₂)
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pfm₁pfm₂⊆pfm₁fm₂ k' v' k',v'∈pfm₁pfm₂
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with ((v₁ , v₂) , (refl , (k,v₁∈pfm₁ , k,v₂∈pfm₂))) ← Provenance-union (pop fm₁) (pop fm₂) k',v'∈pfm₁pfm₂
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with ((v₁ , v₂) , (refl , (k,v₁∈pfm₁ , k,v₂∈pfm₂)))
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← Provenance-union (pop fm₁) (pop fm₂) k',v'∈pfm₁pfm₂
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with (k≢k' , k',v₁∈fm₁) ← k,v∈pop⇒k,v∈ fm₁ k,v₁∈pfm₁
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with (_ , k',v₂∈fm₂) ← k,v∈pop⇒k,v∈ fm₂ k,v₂∈pfm₂
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= (v₁ ⊔₂ v₂ , (IsLattice.≈-refl lB , k,v∈⇒k,v∈pop (fm₁ ⊔ᵐ fm₂) k≢k' (⊔-combines {m₁ = m₁} {m₂ = m₂} k',v₁∈fm₁ k',v₂∈fm₂)))
|
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=
|
||||
( v₁ ⊔₂ v₂
|
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, ( IsLattice.≈-refl lB
|
||||
, k,v∈⇒k,v∈pop (fm₁ ⊔ᵐ fm₂) k≢k'
|
||||
(⊔-combines {m₁ = m₁} {m₂ = m₂}
|
||||
k',v₁∈fm₁ k',v₂∈fm₂)
|
||||
)
|
||||
)
|
||||
|
||||
from-rest : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → proj₂ (from fm) ≡ from (pop fm)
|
||||
from-rest : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) →
|
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proj₂ (from fm) ≡ from (pop fm)
|
||||
from-rest (((_ ∷ fm') , push _ ufm') , refl) = refl
|
||||
|
||||
from-preserves-≈ : ∀ {ks : List A} → {fm₁ fm₂ : FiniteMap ks} → fm₁ ≈ᵐ fm₂ → (_≈ⁱᵖ_ {ks}) (from fm₁) (from fm₂)
|
||||
from-preserves-≈ {[]} {([] , _) , _} {([] , _) , _} _ = IsEquivalence.≈-refl ≈ᵘ-equiv
|
||||
from-preserves-≈ : ∀ {ks : List A} → {fm₁ fm₂ : FiniteMap ks} →
|
||||
fm₁ ≈ᵐ fm₂ → (_≈ⁱᵖ_ {length ks}) (from fm₁) (from fm₂)
|
||||
from-preserves-≈ {[]} {_} {_} _ = IsEquivalence.≈-refl ≈ᵘ-equiv
|
||||
from-preserves-≈ {k ∷ ks'} {fm₁@(m₁ , _)} {fm₂@(m₂ , _)} fm₁≈fm₂@(kvs₁⊆kvs₂ , kvs₂⊆kvs₁)
|
||||
with first-key-in-map fm₁ | first-key-in-map fm₂ | from-first-value fm₁ | from-first-value fm₂
|
||||
with first-key-in-map fm₁
|
||||
| first-key-in-map fm₂
|
||||
| from-first-value fm₁
|
||||
| from-first-value fm₂
|
||||
... | (v₁ , k,v₁∈fm₁) | (v₂ , k,v₂∈fm₂) | refl | refl
|
||||
with kvs₁⊆kvs₂ _ _ k,v₁∈fm₁
|
||||
... | (v₁' , (v₁≈v₁' , k,v₁'∈fm₂))
|
||||
rewrite Map-functional {m = m₂} k,v₂∈fm₂ k,v₁'∈fm₂
|
||||
rewrite from-rest fm₁ rewrite from-rest fm₂
|
||||
= (v₁≈v₁' , from-preserves-≈ {ks'} {pop fm₁} {pop fm₂} (pop-≈ fm₁ fm₂ fm₁≈fm₂))
|
||||
=
|
||||
( v₁≈v₁'
|
||||
, from-preserves-≈ {ks'} {pop fm₁} {pop fm₂}
|
||||
(pop-≈ fm₁ fm₂ fm₁≈fm₂)
|
||||
)
|
||||
|
||||
to-preserves-≈ : ∀ {ks : List A} (uks : Unique ks) {ip₁ ip₂ : IterProd (length ks)} → _≈ⁱᵖ_ {ks} ip₁ ip₂ → to uks ip₁ ≈ᵐ to uks ip₂
|
||||
to-preserves-≈ : ∀ {ks : List A} (uks : Unique ks) {ip₁ ip₂ : IterProd (length ks)} →
|
||||
_≈ⁱᵖ_ {length ks} ip₁ ip₂ → to uks ip₁ ≈ᵐ to uks ip₂
|
||||
to-preserves-≈ {[]} empty {tt} {tt} _ = ((λ k v ()), (λ k v ()))
|
||||
to-preserves-≈ {k ∷ ks'} uks@(push k≢ks' uks') {ip₁@(v₁ , rest₁)} {ip₂@(v₂ , rest₂)} (v₁≈v₂ , rest₁≈rest₂) = (fm₁⊆fm₂ , fm₂⊆fm₁)
|
||||
where
|
||||
fm₁⊆fm₂ : to uks ip₁ ⊆ᵐ to uks ip₂
|
||||
fm₁⊆fm₂ k v k,v∈kvs₁
|
||||
inductive-step : ∀ {v₁ v₂ : B} {rest₁ rest₂ : IterProd (length ks')} →
|
||||
v₁ ≈₂ v₂ → _≈ⁱᵖ_ {length ks'} rest₁ rest₂ →
|
||||
to uks (v₁ , rest₁) ⊆ᵐ to uks (v₂ , rest₂)
|
||||
inductive-step {v₁} {v₂} {rest₁} {rest₂} v₁≈v₂ rest₁≈rest₂ k v k,v∈kvs₁
|
||||
with ((fm'₁ , ufm'₁) , fm'₁≡ks') ← to uks' rest₁ in p₁
|
||||
with ((fm'₂ , ufm'₂) , fm'₂≡ks') ← to uks' rest₂ in p₂
|
||||
with k,v∈kvs₁
|
||||
... | here refl = (v₂ , (v₁≈v₂ , here refl))
|
||||
... | there k,v∈fm'₁ with refl ← p₁ with refl ← p₂ = let (v' , (v≈v' , k,v'∈kvs₁)) = proj₁ (to-preserves-≈ uks' {rest₁} {rest₂} rest₁≈rest₂) k v k,v∈fm'₁ in (v' , (v≈v' , there k,v'∈kvs₁))
|
||||
... | there k,v∈fm'₁ with refl ← p₁ with refl ← p₂ =
|
||||
let
|
||||
(fm'₁⊆fm'₂ , _) = to-preserves-≈ uks' {rest₁} {rest₂}
|
||||
rest₁≈rest₂
|
||||
(v' , (v≈v' , k,v'∈kvs₁)) = fm'₁⊆fm'₂ k v k,v∈fm'₁
|
||||
in
|
||||
(v' , (v≈v' , there k,v'∈kvs₁))
|
||||
|
||||
fm₁⊆fm₂ : to uks ip₁ ⊆ᵐ to uks ip₂
|
||||
fm₁⊆fm₂ = inductive-step v₁≈v₂ rest₁≈rest₂
|
||||
|
||||
fm₂⊆fm₁ : to uks ip₂ ⊆ᵐ to uks ip₁
|
||||
fm₂⊆fm₁ k v k,v∈kvs₂
|
||||
with ((fm'₁ , ufm'₁) , fm'₁≡ks') ← to uks' rest₁ in p₁
|
||||
with ((fm'₂ , ufm'₂) , fm'₂≡ks') ← to uks' rest₂ in p₂
|
||||
with k,v∈kvs₂
|
||||
... | here refl = (v₁ , (IsLattice.≈-sym lB v₁≈v₂ , here refl))
|
||||
... | there k,v∈fm'₂ with refl ← p₁ with refl ← p₂ = let (v' , (v≈v' , k,v'∈kvs₂)) = proj₂ (to-preserves-≈ uks' {rest₁} {rest₂} rest₁≈rest₂) k v k,v∈fm'₂ in (v' , (v≈v' , there k,v'∈kvs₂))
|
||||
fm₂⊆fm₁ = inductive-step (≈₂-sym v₁≈v₂)
|
||||
(IP.≈-sym (length ks') rest₁≈rest₂)
|
||||
|
||||
from-⊔-distr : ∀ {ks : List A} → (fm₁ fm₂ : FiniteMap ks) → _≈ⁱᵖ_ {ks} (from (fm₁ ⊔ᵐ fm₂)) (_⊔ⁱᵖ_ {ks} (from fm₁) (from fm₂))
|
||||
from-⊔-distr : ∀ {ks : List A} → (fm₁ fm₂ : FiniteMap ks) → _≈ⁱᵖ_ {length ks} (from (fm₁ ⊔ᵐ fm₂)) (_⊔ⁱᵖ_ {ks} (from fm₁) (from fm₂))
|
||||
from-⊔-distr {[]} fm₁ fm₂ = IsEquivalence.≈-refl ≈ᵘ-equiv
|
||||
from-⊔-distr {k ∷ ks} fm₁@(m₁ , _) fm₂@(m₂ , _)
|
||||
with first-key-in-map (fm₁ ⊔ᵐ fm₂) | first-key-in-map fm₁ | first-key-in-map fm₂ | from-first-value (fm₁ ⊔ᵐ fm₂) | from-first-value fm₁ | from-first-value fm₂
|
||||
|
|
|
|||
Loading…
Reference in New Issue