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MatthewDaggitt merged 3 commits into from
Mar 12, 2024
Merged

Some properties of upTo and downFrom #2316

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65b766b
Some properties of upTo and downFrom
Taneb Mar 12, 2024
001bc7a
Rename things per review comments
Taneb Mar 12, 2024
8c7cced
Fix changelog typo
Taneb Mar 12, 2024
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12 changes: 12 additions & 0 deletions CHANGELOG.md
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Original file line number Diff line number Diff line change
Expand Up @@ -140,6 +140,18 @@ Additions to existing modules
pattern divides k eq = Data.Nat.Divisibility.divides k eq
```

* In `Data.List.Properties`:
```agda
applyUpTo-∷ʳ : applyUpTo f n ∷ʳ f n ≡ applyUpTo f (suc n)
applyDownFrom-∷ʳ : applyDownFrom (f ∘ suc) n ∷ʳ f 0 ≡ applyDownFrom f (suc n)
upTo-∷ʳ : upTo n ∷ʳ n ≡ upTo (suc n)
downFrom-∷ʳ : applyDownFrom suc n ∷ʳ 0 ≡ downFrom (suc n)
applyUpTo-applyDownFrom : reverse (applyUpTo f n) ≡ applyDownFrom f n
upTo-downFrom : reverse (upTo n) ≡ downFrom n
applyDownFrom-applyUpTo : reverse (applyDownFrom f n) ≡ applyUpTo f n
downFrom-upTo : reverse (downFrom n) ≡ upTo n
```

* In `Data.List.Relation.Unary.All.Properties`:
```agda
All-catMaybes⁺ : All (Maybe.All P) xs → All P (catMaybes xs)
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41 changes: 40 additions & 1 deletion src/Data/List/Properties.agda
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Expand Up @@ -657,8 +657,12 @@ lookup-applyUpTo : ∀ (f : ℕ → A) n i → lookup (applyUpTo f n) i ≡ f (t
lookup-applyUpTo f (suc n) zero = refl
lookup-applyUpTo f (suc n) (suc i) = lookup-applyUpTo (f ∘ suc) n i

applyUpTo-∷ʳ : ∀ (f : ℕ → A) n → applyUpTo f n ∷ʳ f n ≡ applyUpTo f (suc n)
applyUpTo-∷ʳ f zero = refl
applyUpTo-∷ʳ f (suc n) = cong (f 0 ∷_) (applyUpTo-∷ʳ (f ∘ suc) n)

------------------------------------------------------------------------
-- applyUpTo
-- applyDownFrom

module _ (f : ℕ → A) where

Expand All @@ -670,6 +674,10 @@ module _ (f : ℕ → A) where
lookup-applyDownFrom (suc n) zero = refl
lookup-applyDownFrom (suc n) (suc i) = lookup-applyDownFrom n i

applyDownFrom-∷ʳ : ∀ n → applyDownFrom (f ∘ suc) n ∷ʳ f 0 ≡ applyDownFrom f (suc n)
applyDownFrom-∷ʳ zero = refl
applyDownFrom-∷ʳ (suc n) = cong (f (suc n) ∷_) (applyDownFrom-∷ʳ n)

------------------------------------------------------------------------
-- upTo

Expand All @@ -679,6 +687,9 @@ length-upTo = length-applyUpTo id
lookup-upTo : ∀ n i → lookup (upTo n) i ≡ toℕ i
lookup-upTo = lookup-applyUpTo id

upTo-∷ʳ : ∀ n → upTo n ∷ʳ n ≡ upTo (suc n)
upTo-∷ʳ = applyUpTo-∷ʳ id

------------------------------------------------------------------------
-- downFrom

Expand All @@ -688,6 +699,9 @@ length-downFrom = length-applyDownFrom id
lookup-downFrom : ∀ n i → lookup (downFrom n) i ≡ n ∸ (suc (toℕ i))
lookup-downFrom = lookup-applyDownFrom id

downFrom-∷ʳ : ∀ n → applyDownFrom suc n ∷ʳ 0 ≡ downFrom (suc n)
downFrom-∷ʳ = applyDownFrom-∷ʳ id

------------------------------------------------------------------------
-- tabulate

Expand Down Expand Up @@ -1173,6 +1187,31 @@ reverse-foldl : ∀ (f : B → A → B) b xs →
foldl f b (reverse xs) ≡ foldr (flip f) b xs
reverse-foldl f b xs = foldl-ʳ++ f b xs

------------------------------------------------------------------------
-- reverse, applyUpTo, and applyDownFrom

applyUpTo-applyDownFrom : ∀ (f : ℕ → A) n → reverse (applyUpTo f n) ≡ applyDownFrom f n
applyUpTo-applyDownFrom f zero = refl
applyUpTo-applyDownFrom f (suc n) = begin
reverse (f 0 ∷ applyUpTo (f ∘ suc) n) ≡⟨ reverse-++ [ f 0 ] (applyUpTo (f ∘ suc) n) ⟩
reverse (applyUpTo (f ∘ suc) n) ∷ʳ f 0 ≡⟨ cong (_∷ʳ f 0) (applyUpTo-applyDownFrom (f ∘ suc) n) ⟩
applyDownFrom (f ∘ suc) n ∷ʳ f 0 ≡⟨ applyDownFrom-∷ʳ f n ⟩
applyDownFrom f (suc n) ∎

upTo-downFrom : ∀ n → reverse (upTo n) ≡ downFrom n
upTo-downFrom = applyUpTo-applyDownFrom id

applyDownFrom-applyUpTo : ∀ (f : ℕ → A) n → reverse (applyDownFrom f n) ≡ applyUpTo f n
applyDownFrom-applyUpTo f zero = refl
applyDownFrom-applyUpTo f (suc n) = begin
reverse (f n ∷ applyDownFrom f n) ≡⟨ reverse-++ [ f n ] (applyDownFrom f n) ⟩
reverse (applyDownFrom f n) ∷ʳ f n ≡⟨ cong (_∷ʳ f n) (applyDownFrom-applyUpTo f n) ⟩
applyUpTo f n ∷ʳ f n ≡⟨ applyUpTo-∷ʳ f n ⟩
applyUpTo f (suc n) ∎

downFrom-upTo : ∀ n → reverse (downFrom n) ≡ upTo n
downFrom-upTo = applyDownFrom-applyUpTo id

------------------------------------------------------------------------
-- _∷ʳ_

Expand Down