------------------------------------------------------------------------ -- The Agda standard library -- -- List-related properties ------------------------------------------------------------------------ -- Note that the lemmas below could be generalised to work with other -- equalities than _≡_. {-# OPTIONS --without-K --safe #-} module Data.List.Properties where open import Algebra.Bundles open import Algebra.Definitions as AlgebraicDefinitions using (Involutive) import Algebra.Structures as AlgebraicStructures open import Data.Bool.Base using (Bool; false; true; not; if_then_else_) open import Data.Fin.Base using (Fin; zero; suc; cast; toℕ; inject₁) open import Data.List.Base as List open import Data.List.Membership.Propositional using (_∈_) open import Data.List.Relation.Unary.All using (All; []; _∷_) open import Data.List.Relation.Unary.Any using (Any; here; there) open import Data.Maybe.Base using (Maybe; just; nothing) open import Data.Nat.Base open import Data.Nat.Divisibility open import Data.Nat.Properties open import Data.Product as Prod hiding (map; zip) import Data.Product.Relation.Unary.All as Prod using (All) open import Data.Sum.Base using (_⊎_; inj₁; inj₂) open import Data.These.Base as These using (These; this; that; these) open import Function open import Level using (Level) open import Relation.Binary as B using (DecidableEquality) import Relation.Binary.Reasoning.Setoid as EqR open import Relation.Binary.PropositionalEquality as P hiding ([_]) open import Relation.Binary as B using (Rel) open import Relation.Nullary.Reflects using (invert) open import Relation.Nullary using (¬_; Dec; does; _because_; yes; no) open import Relation.Nullary.Negation using (contradiction; ¬?) open import Relation.Nullary.Decidable as Decidable using (isYes; map′; ⌊_⌋) open import Relation.Nullary.Product using (_×-dec_) open import Relation.Unary using (Pred; Decidable; ∁) open import Relation.Unary.Properties using (∁?) open ≡-Reasoning private variable a b c d e p : Level A : Set a B : Set b C : Set c D : Set d E : Set e ----------------------------------------------------------------------- -- _∷_ module _ {x y : A} {xs ys : List A} where ∷-injective : x ∷ xs ≡ y List.∷ ys → x ≡ y × xs ≡ ys ∷-injective refl = (refl , refl) ∷-injectiveˡ : x ∷ xs ≡ y List.∷ ys → x ≡ y ∷-injectiveˡ refl = refl ∷-injectiveʳ : x ∷ xs ≡ y List.∷ ys → xs ≡ ys ∷-injectiveʳ refl = refl ∷-dec : Dec (x ≡ y) → Dec (xs ≡ ys) → Dec (x List.∷ xs ≡ y ∷ ys) ∷-dec x≟y xs≟ys = Decidable.map′ (uncurry (cong₂ _∷_)) ∷-injective (x≟y ×-dec xs≟ys) ≡-dec : DecidableEquality A → DecidableEquality (List A) ≡-dec _≟_ [] [] = yes refl ≡-dec _≟_ (x ∷ xs) [] = no λ() ≡-dec _≟_ [] (y ∷ ys) = no λ() ≡-dec _≟_ (x ∷ xs) (y ∷ ys) = ∷-dec (x ≟ y) (≡-dec _≟_ xs ys) ------------------------------------------------------------------------ -- map map-id : map id ≗ id {A = List A} map-id [] = refl map-id (x ∷ xs) = cong (x ∷_) (map-id xs) map-id₂ : ∀ {f : A → A} {xs} → All (λ x → f x ≡ x) xs → map f xs ≡ xs map-id₂ [] = refl map-id₂ (fx≡x ∷ pxs) = cong₂ _∷_ fx≡x (map-id₂ pxs) map-++-commute : ∀ (f : A → B) xs ys → map f (xs ++ ys) ≡ map f xs ++ map f ys map-++-commute f [] ys = refl map-++-commute f (x ∷ xs) ys = cong (f x ∷_) (map-++-commute f xs ys) map-cong : ∀ {f g : A → B} → f ≗ g → map f ≗ map g map-cong f≗g [] = refl map-cong f≗g (x ∷ xs) = cong₂ _∷_ (f≗g x) (map-cong f≗g xs) map-cong₂ : ∀ {f g : A → B} {xs} → All (λ x → f x ≡ g x) xs → map f xs ≡ map g xs map-cong₂ [] = refl map-cong₂ (fx≡gx ∷ fxs≡gxs) = cong₂ _∷_ fx≡gx (map-cong₂ fxs≡gxs) length-map : ∀ (f : A → B) xs → length (map f xs) ≡ length xs length-map f [] = refl length-map f (x ∷ xs) = cong suc (length-map f xs) map-compose : {g : B → C} {f : A → B} → map (g ∘ f) ≗ map g ∘ map f map-compose [] = refl map-compose (x ∷ xs) = cong (_ ∷_) (map-compose xs) map-injective : ∀ {f : A → B} → Injective _≡_ _≡_ f → Injective _≡_ _≡_ (map f) map-injective finj {[]} {[]} eq = refl map-injective finj {x ∷ xs} {y ∷ ys} eq = let fx≡fy , fxs≡fys = ∷-injective eq in cong₂ _∷_ (finj fx≡fy) (map-injective finj fxs≡fys) ------------------------------------------------------------------------ -- mapMaybe mapMaybe-just : (xs : List A) → mapMaybe just xs ≡ xs mapMaybe-just [] = refl mapMaybe-just (x ∷ xs) = cong (x ∷_) (mapMaybe-just xs) mapMaybe-nothing : (xs : List A) → mapMaybe {B = A} (λ _ → nothing) xs ≡ [] mapMaybe-nothing [] = refl mapMaybe-nothing (x ∷ xs) = mapMaybe-nothing xs module _ (f : A → Maybe B) where mapMaybe-concatMap : mapMaybe f ≗ concatMap (fromMaybe ∘ f) mapMaybe-concatMap [] = refl mapMaybe-concatMap (x ∷ xs) with f x ... | just y = cong (y ∷_) (mapMaybe-concatMap xs) ... | nothing = mapMaybe-concatMap xs length-mapMaybe : ∀ xs → length (mapMaybe f xs) ≤ length xs length-mapMaybe [] = z≤n length-mapMaybe (x ∷ xs) with f x ... | just y = s≤s (length-mapMaybe xs) ... | nothing = ≤-step (length-mapMaybe xs) ------------------------------------------------------------------------ -- _++_ length-++ : ∀ (xs : List A) {ys} → length (xs ++ ys) ≡ length xs + length ys length-++ [] = refl length-++ (x ∷ xs) = cong suc (length-++ xs) module _ {A : Set a} where open AlgebraicDefinitions {A = List A} _≡_ open AlgebraicStructures {A = List A} _≡_ ++-assoc : Associative _++_ ++-assoc [] ys zs = refl ++-assoc (x ∷ xs) ys zs = cong (x ∷_) (++-assoc xs ys zs) ++-identityˡ : LeftIdentity [] _++_ ++-identityˡ xs = refl ++-identityʳ : RightIdentity [] _++_ ++-identityʳ [] = refl ++-identityʳ (x ∷ xs) = cong (x ∷_) (++-identityʳ xs) ++-identity : Identity [] _++_ ++-identity = ++-identityˡ , ++-identityʳ ++-identityʳ-unique : ∀ (xs : List A) {ys} → xs ≡ xs ++ ys → ys ≡ [] ++-identityʳ-unique [] refl = refl ++-identityʳ-unique (x ∷ xs) eq = ++-identityʳ-unique xs (proj₂ (∷-injective eq)) ++-identityˡ-unique : ∀ {xs} (ys : List A) → xs ≡ ys ++ xs → ys ≡ [] ++-identityˡ-unique [] _ = refl ++-identityˡ-unique {xs = x ∷ xs} (y ∷ ys) eq with ++-identityˡ-unique (ys ++ [ x ]) (begin xs ≡⟨ proj₂ (∷-injective eq) ⟩ ys ++ x ∷ xs ≡⟨ sym (++-assoc ys [ x ] xs) ⟩ (ys ++ [ x ]) ++ xs ∎) ++-identityˡ-unique {xs = x ∷ xs} (y ∷ [] ) eq | () ++-identityˡ-unique {xs = x ∷ xs} (y ∷ _ ∷ _) eq | () ++-cancelˡ : ∀ xs {ys zs : List A} → xs ++ ys ≡ xs ++ zs → ys ≡ zs ++-cancelˡ [] ys≡zs = ys≡zs ++-cancelˡ (x ∷ xs) x∷xs++ys≡x∷xs++zs = ++-cancelˡ xs (∷-injectiveʳ x∷xs++ys≡x∷xs++zs) ++-cancelʳ : ∀ {xs : List A} ys zs → ys ++ xs ≡ zs ++ xs → ys ≡ zs ++-cancelʳ {_} [] [] _ = refl ++-cancelʳ {xs} [] (z ∷ zs) eq = contradiction (trans (cong length eq) (length-++ (z ∷ zs))) (m≢1+n+m (length xs)) ++-cancelʳ {xs} (y ∷ ys) [] eq = contradiction (trans (sym (length-++ (y ∷ ys))) (cong length eq)) (m≢1+n+m (length xs) ∘ sym) ++-cancelʳ {_} (y ∷ ys) (z ∷ zs) eq = cong₂ _∷_ (∷-injectiveˡ eq) (++-cancelʳ ys zs (∷-injectiveʳ eq)) ++-cancel : Cancellative _++_ ++-cancel = ++-cancelˡ , ++-cancelʳ ++-conicalˡ : ∀ (xs ys : List A) → xs ++ ys ≡ [] → xs ≡ [] ++-conicalˡ [] _ refl = refl ++-conicalʳ : ∀ (xs ys : List A) → xs ++ ys ≡ [] → ys ≡ [] ++-conicalʳ [] _ refl = refl ++-conical : Conical [] _++_ ++-conical = ++-conicalˡ , ++-conicalʳ ++-isMagma : IsMagma _++_ ++-isMagma = record { isEquivalence = isEquivalence ; ∙-cong = cong₂ _++_ } ++-isSemigroup : IsSemigroup _++_ ++-isSemigroup = record { isMagma = ++-isMagma ; assoc = ++-assoc } ++-isMonoid : IsMonoid _++_ [] ++-isMonoid = record { isSemigroup = ++-isSemigroup ; identity = ++-identity } module _ (A : Set a) where ++-semigroup : Semigroup a a ++-semigroup = record { Carrier = List A ; isSemigroup = ++-isSemigroup } ++-monoid : Monoid a a ++-monoid = record { Carrier = List A ; isMonoid = ++-isMonoid } ------------------------------------------------------------------------ -- alignWith module _ {f g : These A B → C} where alignWith-cong : f ≗ g → ∀ as → alignWith f as ≗ alignWith g as alignWith-cong f≗g [] bs = map-cong (f≗g ∘ that) bs alignWith-cong f≗g as@(_ ∷ _) [] = map-cong (f≗g ∘ this) as alignWith-cong f≗g (a ∷ as) (b ∷ bs) = cong₂ _∷_ (f≗g (these a b)) (alignWith-cong f≗g as bs) length-alignWith : ∀ xs ys → length (alignWith f xs ys) ≡ length xs ⊔ length ys length-alignWith [] ys = length-map (f ∘′ that) ys length-alignWith xs@(_ ∷ _) [] = length-map (f ∘′ this) xs length-alignWith (x ∷ xs) (y ∷ ys) = cong suc (length-alignWith xs ys) alignWith-map : (g : D → A) (h : E → B) → ∀ xs ys → alignWith f (map g xs) (map h ys) ≡ alignWith (f ∘′ These.map g h) xs ys alignWith-map g h [] ys = sym (map-compose ys) alignWith-map g h xs@(_ ∷ _) [] = sym (map-compose xs) alignWith-map g h (x ∷ xs) (y ∷ ys) = cong₂ _∷_ refl (alignWith-map g h xs ys) map-alignWith : ∀ (g : C → D) → ∀ xs ys → map g (alignWith f xs ys) ≡ alignWith (g ∘′ f) xs ys map-alignWith g [] ys = sym (map-compose ys) map-alignWith g xs@(_ ∷ _) [] = sym (map-compose xs) map-alignWith g (x ∷ xs) (y ∷ ys) = cong₂ _∷_ refl (map-alignWith g xs ys) ------------------------------------------------------------------------ -- zipWith module _ (f : A → A → B) where zipWith-comm : (∀ x y → f x y ≡ f y x) → ∀ xs ys → zipWith f xs ys ≡ zipWith f ys xs zipWith-comm f-comm [] [] = refl zipWith-comm f-comm [] (x ∷ ys) = refl zipWith-comm f-comm (x ∷ xs) [] = refl zipWith-comm f-comm (x ∷ xs) (y ∷ ys) = cong₂ _∷_ (f-comm x y) (zipWith-comm f-comm xs ys) module _ (f : A → B → C) where zipWith-zeroˡ : ∀ xs → zipWith f [] xs ≡ [] zipWith-zeroˡ [] = refl zipWith-zeroˡ (x ∷ xs) = refl zipWith-zeroʳ : ∀ xs → zipWith f xs [] ≡ [] zipWith-zeroʳ [] = refl zipWith-zeroʳ (x ∷ xs) = refl length-zipWith : ∀ xs ys → length (zipWith f xs ys) ≡ length xs ⊓ length ys length-zipWith [] [] = refl length-zipWith [] (y ∷ ys) = refl length-zipWith (x ∷ xs) [] = refl length-zipWith (x ∷ xs) (y ∷ ys) = cong suc (length-zipWith xs ys) zipWith-map : ∀ {d e} {D : Set d} {E : Set e} (g : D → A) (h : E → B) → ∀ xs ys → zipWith f (map g xs) (map h ys) ≡ zipWith (λ x y → f (g x) (h y)) xs ys zipWith-map g h [] [] = refl zipWith-map g h [] (y ∷ ys) = refl zipWith-map g h (x ∷ xs) [] = refl zipWith-map g h (x ∷ xs) (y ∷ ys) = cong₂ _∷_ refl (zipWith-map g h xs ys) map-zipWith : ∀ {d} {D : Set d} (g : C → D) → ∀ xs ys → map g (zipWith f xs ys) ≡ zipWith (λ x y → g (f x y)) xs ys map-zipWith g [] [] = refl map-zipWith g [] (y ∷ ys) = refl map-zipWith g (x ∷ xs) [] = refl map-zipWith g (x ∷ xs) (y ∷ ys) = cong₂ _∷_ refl (map-zipWith g xs ys) ------------------------------------------------------------------------ -- unalignWith unalignWith-this : unalignWith ((A → These A B) ∋ this) ≗ (_, []) unalignWith-this [] = refl unalignWith-this (a ∷ as) = cong (Prod.map₁ (a ∷_)) (unalignWith-this as) unalignWith-that : unalignWith ((B → These A B) ∋ that) ≗ ([] ,_) unalignWith-that [] = refl unalignWith-that (b ∷ bs) = cong (Prod.map₂ (b ∷_)) (unalignWith-that bs) module _ {f g : C → These A B} where unalignWith-cong : f ≗ g → unalignWith f ≗ unalignWith g unalignWith-cong f≗g [] = refl unalignWith-cong f≗g (c ∷ cs) with f c | g c | f≗g c ... | this a | ._ | refl = cong (Prod.map₁ (a ∷_)) (unalignWith-cong f≗g cs) ... | that b | ._ | refl = cong (Prod.map₂ (b ∷_)) (unalignWith-cong f≗g cs) ... | these a b | ._ | refl = cong (Prod.map (a ∷_) (b ∷_)) (unalignWith-cong f≗g cs) module _ (f : C → These A B) where unalignWith-map : (g : D → C) → ∀ ds → unalignWith f (map g ds) ≡ unalignWith (f ∘′ g) ds unalignWith-map g [] = refl unalignWith-map g (d ∷ ds) with f (g d) ... | this a = cong (Prod.map₁ (a ∷_)) (unalignWith-map g ds) ... | that b = cong (Prod.map₂ (b ∷_)) (unalignWith-map g ds) ... | these a b = cong (Prod.map (a ∷_) (b ∷_)) (unalignWith-map g ds) map-unalignWith : (g : A → D) (h : B → E) → Prod.map (map g) (map h) ∘′ unalignWith f ≗ unalignWith (These.map g h ∘′ f) map-unalignWith g h [] = refl map-unalignWith g h (c ∷ cs) with f c ... | this a = cong (Prod.map₁ (g a ∷_)) (map-unalignWith g h cs) ... | that b = cong (Prod.map₂ (h b ∷_)) (map-unalignWith g h cs) ... | these a b = cong (Prod.map (g a ∷_) (h b ∷_)) (map-unalignWith g h cs) unalignWith-alignWith : (g : These A B → C) → f ∘′ g ≗ id → ∀ as bs → unalignWith f (alignWith g as bs) ≡ (as , bs) unalignWith-alignWith g g∘f≗id [] bs = begin unalignWith f (map (g ∘′ that) bs) ≡⟨ unalignWith-map (g ∘′ that) bs ⟩ unalignWith (f ∘′ g ∘′ that) bs ≡⟨ unalignWith-cong (g∘f≗id ∘ that) bs ⟩ unalignWith that bs ≡⟨ unalignWith-that bs ⟩ [] , bs ∎ unalignWith-alignWith g g∘f≗id as@(_ ∷ _) [] = begin unalignWith f (map (g ∘′ this) as) ≡⟨ unalignWith-map (g ∘′ this) as ⟩ unalignWith (f ∘′ g ∘′ this) as ≡⟨ unalignWith-cong (g∘f≗id ∘ this) as ⟩ unalignWith this as ≡⟨ unalignWith-this as ⟩ as , [] ∎ unalignWith-alignWith g g∘f≗id (a ∷ as) (b ∷ bs) rewrite g∘f≗id (these a b) = cong (Prod.map (a ∷_) (b ∷_)) (unalignWith-alignWith g g∘f≗id as bs) ------------------------------------------------------------------------ -- unzipWith module _ (f : A → B × C) where length-unzipWith₁ : ∀ xys → length (proj₁ (unzipWith f xys)) ≡ length xys length-unzipWith₁ [] = refl length-unzipWith₁ (x ∷ xys) = cong suc (length-unzipWith₁ xys) length-unzipWith₂ : ∀ xys → length (proj₂ (unzipWith f xys)) ≡ length xys length-unzipWith₂ [] = refl length-unzipWith₂ (x ∷ xys) = cong suc (length-unzipWith₂ xys) zipWith-unzipWith : (g : B → C → A) → uncurry′ g ∘ f ≗ id → uncurry′ (zipWith g) ∘ (unzipWith f) ≗ id zipWith-unzipWith g f∘g≗id [] = refl zipWith-unzipWith g f∘g≗id (x ∷ xs) = cong₂ _∷_ (f∘g≗id x) (zipWith-unzipWith g f∘g≗id xs) ------------------------------------------------------------------------ -- foldr foldr-universal : ∀ (h : List A → B) f e → (h [] ≡ e) → (∀ x xs → h (x ∷ xs) ≡ f x (h xs)) → h ≗ foldr f e foldr-universal h f e base step [] = base foldr-universal h f e base step (x ∷ xs) = begin h (x ∷ xs) ≡⟨ step x xs ⟩ f x (h xs) ≡⟨ cong (f x) (foldr-universal h f e base step xs) ⟩ f x (foldr f e xs) ∎ foldr-cong : ∀ {f g : A → B → B} {d e : B} → (∀ x y → f x y ≡ g x y) → d ≡ e → foldr f d ≗ foldr g e foldr-cong f≗g refl [] = refl foldr-cong f≗g d≡e (x ∷ xs) rewrite foldr-cong f≗g d≡e xs = f≗g x _ foldr-fusion : ∀ (h : B → C) {f : A → B → B} {g : A → C → C} (e : B) → (∀ x y → h (f x y) ≡ g x (h y)) → h ∘ foldr f e ≗ foldr g (h e) foldr-fusion h {f} {g} e fuse = foldr-universal (h ∘ foldr f e) g (h e) refl (λ x xs → fuse x (foldr f e xs)) id-is-foldr : id {A = List A} ≗ foldr _∷_ [] id-is-foldr = foldr-universal id _∷_ [] refl (λ _ _ → refl) ++-is-foldr : (xs ys : List A) → xs ++ ys ≡ foldr _∷_ ys xs ++-is-foldr xs ys = begin xs ++ ys ≡⟨ cong (_++ ys) (id-is-foldr xs) ⟩ foldr _∷_ [] xs ++ ys ≡⟨ foldr-fusion (_++ ys) [] (λ _ _ → refl) xs ⟩ foldr _∷_ ([] ++ ys) xs ≡⟨⟩ foldr _∷_ ys xs ∎ foldr-++ : ∀ (f : A → B → B) x ys zs → foldr f x (ys ++ zs) ≡ foldr f (foldr f x zs) ys foldr-++ f x [] zs = refl foldr-++ f x (y ∷ ys) zs = cong (f y) (foldr-++ f x ys zs) map-is-foldr : {f : A → B} → map f ≗ foldr (λ x ys → f x ∷ ys) [] map-is-foldr {f = f} xs = begin map f xs ≡⟨ cong (map f) (id-is-foldr xs) ⟩ map f (foldr _∷_ [] xs) ≡⟨ foldr-fusion (map f) [] (λ _ _ → refl) xs ⟩ foldr (λ x ys → f x ∷ ys) [] xs ∎ foldr-∷ʳ : ∀ (f : A → B → B) x y ys → foldr f x (ys ∷ʳ y) ≡ foldr f (f y x) ys foldr-∷ʳ f x y [] = refl foldr-∷ʳ f x y (z ∷ ys) = cong (f z) (foldr-∷ʳ f x y ys) -- Interaction with predicates module _ {P : Pred A p} {f : A → A → A} where foldr-forcesᵇ : (∀ x y → P (f x y) → P x × P y) → ∀ e xs → P (foldr f e xs) → All P xs foldr-forcesᵇ _ _ [] _ = [] foldr-forcesᵇ forces _ (x ∷ xs) Pfold with forces _ _ Pfold ... | (px , pfxs) = px ∷ foldr-forcesᵇ forces _ xs pfxs foldr-preservesᵇ : (∀ {x y} → P x → P y → P (f x y)) → ∀ {e xs} → P e → All P xs → P (foldr f e xs) foldr-preservesᵇ _ Pe [] = Pe foldr-preservesᵇ pres Pe (px ∷ pxs) = pres px (foldr-preservesᵇ pres Pe pxs) foldr-preservesʳ : (∀ x {y} → P y → P (f x y)) → ∀ {e} → P e → ∀ xs → P (foldr f e xs) foldr-preservesʳ pres Pe [] = Pe foldr-preservesʳ pres Pe (_ ∷ xs) = pres _ (foldr-preservesʳ pres Pe xs) foldr-preservesᵒ : (∀ x y → P x ⊎ P y → P (f x y)) → ∀ e xs → P e ⊎ Any P xs → P (foldr f e xs) foldr-preservesᵒ pres e [] (inj₁ Pe) = Pe foldr-preservesᵒ pres e (x ∷ xs) (inj₁ Pe) = pres _ _ (inj₂ (foldr-preservesᵒ pres e xs (inj₁ Pe))) foldr-preservesᵒ pres e (x ∷ xs) (inj₂ (here px)) = pres _ _ (inj₁ px) foldr-preservesᵒ pres e (x ∷ xs) (inj₂ (there pxs)) = pres _ _ (inj₂ (foldr-preservesᵒ pres e xs (inj₂ pxs))) ------------------------------------------------------------------------ -- foldl foldl-++ : ∀ (f : A → B → A) x ys zs → foldl f x (ys ++ zs) ≡ foldl f (foldl f x ys) zs foldl-++ f x [] zs = refl foldl-++ f x (y ∷ ys) zs = foldl-++ f (f x y) ys zs foldl-∷ʳ : ∀ (f : A → B → A) x y ys → foldl f x (ys ∷ʳ y) ≡ f (foldl f x ys) y foldl-∷ʳ f x y [] = refl foldl-∷ʳ f x y (z ∷ ys) = foldl-∷ʳ f (f x z) y ys ------------------------------------------------------------------------ -- concat concat-map : ∀ {f : A → B} → concat ∘ map (map f) ≗ map f ∘ concat concat-map {f = f} xss = begin concat (map (map f) xss) ≡⟨ cong concat (map-is-foldr xss) ⟩ concat (foldr (λ xs → map f xs ∷_) [] xss) ≡⟨ foldr-fusion concat [] (λ _ _ → refl) xss ⟩ foldr (λ ys → map f ys ++_) [] xss ≡⟨ sym (foldr-fusion (map f) [] (map-++-commute f) xss) ⟩ map f (concat xss) ∎ concat-++ : (xss yss : List (List A)) → concat xss ++ concat yss ≡ concat (xss ++ yss) concat-++ [] yss = refl concat-++ ([] ∷ xss) yss = concat-++ xss yss concat-++ ((x ∷ xs) ∷ xss) yss = cong (x ∷_) (concat-++ (xs ∷ xss) yss) concat-concat : concat {A = A} ∘ map concat ≗ concat ∘ concat concat-concat [] = refl concat-concat (xss ∷ xsss) = begin concat (map concat (xss ∷ xsss)) ≡⟨ cong (concat xss ++_) (concat-concat xsss) ⟩ concat xss ++ concat (concat xsss) ≡⟨ concat-++ xss (concat xsss) ⟩ concat (concat (xss ∷ xsss)) ∎ concat-[-] : concat {A = A} ∘ map [_] ≗ id concat-[-] [] = refl concat-[-] (x ∷ xs) = cong (x ∷_) (concat-[-] xs) ------------------------------------------------------------------------ -- sum sum-++-commute : ∀ xs ys → sum (xs ++ ys) ≡ sum xs + sum ys sum-++-commute [] ys = refl sum-++-commute (x ∷ xs) ys = begin x + sum (xs ++ ys) ≡⟨ cong (x +_) (sum-++-commute xs ys) ⟩ x + (sum xs + sum ys) ≡⟨ sym (+-assoc x _ _) ⟩ (x + sum xs) + sum ys ∎ ------------------------------------------------------------------------ -- product ∈⇒∣product : ∀ {n ns} → n ∈ ns → n ∣ product ns ∈⇒∣product {n} {n ∷ ns} (here refl) = divides (product ns) (*-comm n (product ns)) ∈⇒∣product {n} {m ∷ ns} (there n∈ns) = ∣n⇒∣m*n m (∈⇒∣product n∈ns) ------------------------------------------------------------------------ -- replicate length-replicate : ∀ n {x : A} → length (replicate n x) ≡ n length-replicate zero = refl length-replicate (suc n) = cong suc (length-replicate n) ------------------------------------------------------------------------ -- scanr scanr-defn : ∀ (f : A → B → B) (e : B) → scanr f e ≗ map (foldr f e) ∘ tails scanr-defn f e [] = refl scanr-defn f e (x ∷ []) = refl scanr-defn f e (x ∷ y ∷ xs) with scanr f e (y ∷ xs) | scanr-defn f e (y ∷ xs) ... | [] | () ... | z ∷ zs | eq with ∷-injective eq ... | z≡fy⦇f⦈xs , _ = cong₂ (λ z → f x z ∷_) z≡fy⦇f⦈xs eq ------------------------------------------------------------------------ -- scanl scanl-defn : ∀ (f : A → B → A) (e : A) → scanl f e ≗ map (foldl f e) ∘ inits scanl-defn f e [] = refl scanl-defn f e (x ∷ xs) = cong (e ∷_) (begin scanl f (f e x) xs ≡⟨ scanl-defn f (f e x) xs ⟩ map (foldl f (f e x)) (inits xs) ≡⟨ refl ⟩ map (foldl f e ∘ (x ∷_)) (inits xs) ≡⟨ map-compose (inits xs) ⟩ map (foldl f e) (map (x ∷_) (inits xs)) ∎) ------------------------------------------------------------------------ -- applyUpTo length-applyUpTo : ∀ (f : ℕ → A) n → length (applyUpTo f n) ≡ n length-applyUpTo f zero = refl length-applyUpTo f (suc n) = cong suc (length-applyUpTo (f ∘ suc) n) lookup-applyUpTo : ∀ (f : ℕ → A) n i → lookup (applyUpTo f n) i ≡ f (toℕ i) lookup-applyUpTo f (suc n) zero = refl lookup-applyUpTo f (suc n) (suc i) = lookup-applyUpTo (f ∘ suc) n i ------------------------------------------------------------------------ -- applyUpTo module _ (f : ℕ → A) where length-applyDownFrom : ∀ n → length (applyDownFrom f n) ≡ n length-applyDownFrom zero = refl length-applyDownFrom (suc n) = cong suc (length-applyDownFrom n) lookup-applyDownFrom : ∀ n i → lookup (applyDownFrom f n) i ≡ f (n ∸ (suc (toℕ i))) lookup-applyDownFrom (suc n) zero = refl lookup-applyDownFrom (suc n) (suc i) = lookup-applyDownFrom n i ------------------------------------------------------------------------ -- upTo length-upTo : ∀ n → length (upTo n) ≡ n length-upTo = length-applyUpTo id lookup-upTo : ∀ n i → lookup (upTo n) i ≡ toℕ i lookup-upTo = lookup-applyUpTo id ------------------------------------------------------------------------ -- downFrom length-downFrom : ∀ n → length (downFrom n) ≡ n length-downFrom = length-applyDownFrom id lookup-downFrom : ∀ n i → lookup (downFrom n) i ≡ n ∸ (suc (toℕ i)) lookup-downFrom = lookup-applyDownFrom id ------------------------------------------------------------------------ -- tabulate tabulate-cong : ∀ {n} {f g : Fin n → A} → f ≗ g → tabulate f ≡ tabulate g tabulate-cong {n = zero} p = refl tabulate-cong {n = suc n} p = cong₂ _∷_ (p zero) (tabulate-cong (p ∘ suc)) tabulate-lookup : ∀ (xs : List A) → tabulate (lookup xs) ≡ xs tabulate-lookup [] = refl tabulate-lookup (x ∷ xs) = cong (_ ∷_) (tabulate-lookup xs) length-tabulate : ∀ {n} → (f : Fin n → A) → length (tabulate f) ≡ n length-tabulate {n = zero} f = refl length-tabulate {n = suc n} f = cong suc (length-tabulate (λ z → f (suc z))) lookup-tabulate : ∀ {n} → (f : Fin n → A) → ∀ i → let i′ = cast (sym (length-tabulate f)) i in lookup (tabulate f) i′ ≡ f i lookup-tabulate f zero = refl lookup-tabulate f (suc i) = lookup-tabulate (f ∘ suc) i map-tabulate : ∀ {n} (g : Fin n → A) (f : A → B) → map f (tabulate g) ≡ tabulate (f ∘ g) map-tabulate {n = zero} g f = refl map-tabulate {n = suc n} g f = cong (_ ∷_) (map-tabulate (g ∘ suc) f) ------------------------------------------------------------------------ -- _[_]%=_ length-%= : ∀ xs k (f : A → A) → length (xs [ k ]%= f) ≡ length xs length-%= (x ∷ xs) zero f = refl length-%= (x ∷ xs) (suc k) f = cong suc (length-%= xs k f) ------------------------------------------------------------------------ -- _[_]∷=_ length-∷= : ∀ xs k (v : A) → length (xs [ k ]∷= v) ≡ length xs length-∷= xs k v = length-%= xs k (const v) map-∷= : ∀ xs k (v : A) (f : A → B) → let eq = sym (length-map f xs) in map f (xs [ k ]∷= v) ≡ map f xs [ cast eq k ]∷= f v map-∷= (x ∷ xs) zero v f = refl map-∷= (x ∷ xs) (suc k) v f = cong (f x ∷_) (map-∷= xs k v f) ------------------------------------------------------------------------ -- _─_ length-─ : ∀ (xs : List A) k → length (xs ─ k) ≡ pred (length xs) length-─ (x ∷ xs) zero = refl length-─ (x ∷ y ∷ xs) (suc k) = cong suc (length-─ (y ∷ xs) k) map-─ : ∀ xs k (f : A → B) → let eq = sym (length-map f xs) in map f (xs ─ k) ≡ map f xs ─ cast eq k map-─ (x ∷ xs) zero f = refl map-─ (x ∷ xs) (suc k) f = cong (f x ∷_) (map-─ xs k f) ------------------------------------------------------------------------ -- take length-take : ∀ n (xs : List A) → length (take n xs) ≡ n ⊓ (length xs) length-take zero xs = refl length-take (suc n) [] = refl length-take (suc n) (x ∷ xs) = cong suc (length-take n xs) ------------------------------------------------------------------------ -- drop length-drop : ∀ n (xs : List A) → length (drop n xs) ≡ length xs ∸ n length-drop zero xs = refl length-drop (suc n) [] = refl length-drop (suc n) (x ∷ xs) = length-drop n xs take++drop : ∀ n (xs : List A) → take n xs ++ drop n xs ≡ xs take++drop zero xs = refl take++drop (suc n) [] = refl take++drop (suc n) (x ∷ xs) = cong (x ∷_) (take++drop n xs) ------------------------------------------------------------------------ -- splitAt splitAt-defn : ∀ n → splitAt {A = A} n ≗ < take n , drop n > splitAt-defn zero xs = refl splitAt-defn (suc n) [] = refl splitAt-defn (suc n) (x ∷ xs) with splitAt n xs | splitAt-defn n xs ... | (ys , zs) | ih = cong (Prod.map (x ∷_) id) ih ------------------------------------------------------------------------ -- takeWhile, dropWhile, and span module _ {P : Pred A p} (P? : Decidable P) where takeWhile++dropWhile : ∀ xs → takeWhile P? xs ++ dropWhile P? xs ≡ xs takeWhile++dropWhile [] = refl takeWhile++dropWhile (x ∷ xs) with does (P? x) ... | true = cong (x ∷_) (takeWhile++dropWhile xs) ... | false = refl span-defn : span P? ≗ < takeWhile P? , dropWhile P? > span-defn [] = refl span-defn (x ∷ xs) with does (P? x) ... | true = cong (Prod.map (x ∷_) id) (span-defn xs) ... | false = refl ------------------------------------------------------------------------ -- filter module _ {P : Pred A p} (P? : Decidable P) where length-filter : ∀ xs → length (filter P? xs) ≤ length xs length-filter [] = z≤n length-filter (x ∷ xs) with does (P? x) ... | false = ≤-step (length-filter xs) ... | true = s≤s (length-filter xs) filter-all : ∀ {xs} → All P xs → filter P? xs ≡ xs filter-all {[]} [] = refl filter-all {x ∷ xs} (px ∷ pxs) with P? x ... | no ¬px = contradiction px ¬px ... | true because _ = cong (x ∷_) (filter-all pxs) filter-notAll : ∀ xs → Any (∁ P) xs → length (filter P? xs) < length xs filter-notAll (x ∷ xs) (here ¬px) with P? x ... | false because _ = s≤s (length-filter xs) ... | yes px = contradiction px ¬px filter-notAll (x ∷ xs) (there any) with does (P? x) ... | false = ≤-step (filter-notAll xs any) ... | true = s≤s (filter-notAll xs any) filter-some : ∀ {xs} → Any P xs → 0 < length (filter P? xs) filter-some {x ∷ xs} (here px) with P? x ... | true because _ = z<s ... | no ¬px = contradiction px ¬px filter-some {x ∷ xs} (there pxs) with does (P? x) ... | true = ≤-step (filter-some pxs) ... | false = filter-some pxs filter-none : ∀ {xs} → All (∁ P) xs → filter P? xs ≡ [] filter-none {[]} [] = refl filter-none {x ∷ xs} (¬px ∷ ¬pxs) with P? x ... | false because _ = filter-none ¬pxs ... | yes px = contradiction px ¬px filter-complete : ∀ {xs} → length (filter P? xs) ≡ length xs → filter P? xs ≡ xs filter-complete {[]} eq = refl filter-complete {x ∷ xs} eq with does (P? x) ... | false = contradiction eq (<⇒≢ (s≤s (length-filter xs))) ... | true = cong (x ∷_) (filter-complete (suc-injective eq)) filter-accept : ∀ {x xs} → P x → filter P? (x ∷ xs) ≡ x ∷ (filter P? xs) filter-accept {x} Px with P? x ... | true because _ = refl ... | no ¬Px = contradiction Px ¬Px filter-reject : ∀ {x xs} → ¬ P x → filter P? (x ∷ xs) ≡ filter P? xs filter-reject {x} ¬Px with P? x ... | yes Px = contradiction Px ¬Px ... | false because _ = refl filter-idem : filter P? ∘ filter P? ≗ filter P? filter-idem [] = refl filter-idem (x ∷ xs) with does (P? x) | inspect does (P? x) ... | false | _ = filter-idem xs ... | true | P.[ eq ] rewrite eq = cong (x ∷_) (filter-idem xs) filter-++ : ∀ xs ys → filter P? (xs ++ ys) ≡ filter P? xs ++ filter P? ys filter-++ [] ys = refl filter-++ (x ∷ xs) ys with does (P? x) ... | true = cong (x ∷_) (filter-++ xs ys) ... | false = filter-++ xs ys ------------------------------------------------------------------------ -- derun and deduplicate module _ {R : Rel A p} (R? : B.Decidable R) where length-derun : ∀ xs → length (derun R? xs) ≤ length xs length-derun [] = ≤-refl length-derun (x ∷ []) = ≤-refl length-derun (x ∷ y ∷ xs) with does (R? x y) | length-derun (y ∷ xs) ... | true | r = ≤-step r ... | false | r = s≤s r length-deduplicate : ∀ xs → length (deduplicate R? xs) ≤ length xs length-deduplicate [] = z≤n length-deduplicate (x ∷ xs) = ≤-begin 1 + length (filter (¬? ∘ R? x) r) ≤⟨ s≤s (length-filter (¬? ∘ R? x) r) ⟩ 1 + length r ≤⟨ s≤s (length-deduplicate xs) ⟩ 1 + length xs ≤-∎ where open ≤-Reasoning renaming (begin_ to ≤-begin_; _∎ to _≤-∎) r = deduplicate R? xs derun-reject : ∀ {x y} xs → R x y → derun R? (x ∷ y ∷ xs) ≡ derun R? (y ∷ xs) derun-reject {x} {y} xs Rxy with R? x y ... | yes _ = refl ... | no ¬Rxy = contradiction Rxy ¬Rxy derun-accept : ∀ {x y} xs → ¬ R x y → derun R? (x ∷ y ∷ xs) ≡ x ∷ derun R? (y ∷ xs) derun-accept {x} {y} xs ¬Rxy with R? x y ... | yes Rxy = contradiction Rxy ¬Rxy ... | no _ = refl ------------------------------------------------------------------------ -- partition module _ {P : Pred A p} (P? : Decidable P) where partition-defn : partition P? ≗ < filter P? , filter (∁? P?) > partition-defn [] = refl partition-defn (x ∷ xs) with does (P? x) ... | true = cong (Prod.map (x ∷_) id) (partition-defn xs) ... | false = cong (Prod.map id (x ∷_)) (partition-defn xs) length-partition : ∀ xs → (let (ys , zs) = partition P? xs) → length ys ≤ length xs × length zs ≤ length xs length-partition [] = z≤n , z≤n length-partition (x ∷ xs) with does (P? x) | length-partition xs ... | true | rec = Prod.map s≤s ≤-step rec ... | false | rec = Prod.map ≤-step s≤s rec ------------------------------------------------------------------------ -- _ʳ++_ ʳ++-defn : ∀ (xs : List A) {ys} → xs ʳ++ ys ≡ reverse xs ++ ys ʳ++-defn [] = refl ʳ++-defn (x ∷ xs) {ys} = begin (x ∷ xs) ʳ++ ys ≡⟨⟩ xs ʳ++ x ∷ ys ≡⟨⟩ xs ʳ++ [ x ] ++ ys ≡⟨ ʳ++-defn xs ⟩ reverse xs ++ [ x ] ++ ys ≡⟨ sym (++-assoc (reverse xs) _ _) ⟩ (reverse xs ++ [ x ]) ++ ys ≡⟨ cong (_++ ys) (sym (ʳ++-defn xs)) ⟩ (xs ʳ++ [ x ]) ++ ys ≡⟨⟩ reverse (x ∷ xs) ++ ys ∎ -- Reverse-append of append is reverse-append after reverse-append. ʳ++-++ : ∀ (xs {ys zs} : List A) → (xs ++ ys) ʳ++ zs ≡ ys ʳ++ xs ʳ++ zs ʳ++-++ [] = refl ʳ++-++ (x ∷ xs) {ys} {zs} = begin (x ∷ xs ++ ys) ʳ++ zs ≡⟨⟩ (xs ++ ys) ʳ++ x ∷ zs ≡⟨ ʳ++-++ xs ⟩ ys ʳ++ xs ʳ++ x ∷ zs ≡⟨⟩ ys ʳ++ (x ∷ xs) ʳ++ zs ∎ -- Reverse-append of reverse-append is commuted reverse-append after append. ʳ++-ʳ++ : ∀ (xs {ys zs} : List A) → (xs ʳ++ ys) ʳ++ zs ≡ ys ʳ++ xs ++ zs ʳ++-ʳ++ [] = refl ʳ++-ʳ++ (x ∷ xs) {ys} {zs} = begin ((x ∷ xs) ʳ++ ys) ʳ++ zs ≡⟨⟩ (xs ʳ++ x ∷ ys) ʳ++ zs ≡⟨ ʳ++-ʳ++ xs ⟩ (x ∷ ys) ʳ++ xs ++ zs ≡⟨⟩ ys ʳ++ (x ∷ xs) ++ zs ∎ -- Length of reverse-append length-ʳ++ : ∀ (xs {ys} : List A) → length (xs ʳ++ ys) ≡ length xs + length ys length-ʳ++ [] = refl length-ʳ++ (x ∷ xs) {ys} = begin length ((x ∷ xs) ʳ++ ys) ≡⟨⟩ length (xs ʳ++ x ∷ ys) ≡⟨ length-ʳ++ xs ⟩ length xs + length (x ∷ ys) ≡⟨ +-suc _ _ ⟩ length (x ∷ xs) + length ys ∎ -- map distributes over reverse-append. map-ʳ++ : (f : A → B) (xs {ys} : List A) → map f (xs ʳ++ ys) ≡ map f xs ʳ++ map f ys map-ʳ++ f [] = refl map-ʳ++ f (x ∷ xs) {ys} = begin map f ((x ∷ xs) ʳ++ ys) ≡⟨⟩ map f (xs ʳ++ x ∷ ys) ≡⟨ map-ʳ++ f xs ⟩ map f xs ʳ++ map f (x ∷ ys) ≡⟨⟩ map f xs ʳ++ f x ∷ map f ys ≡⟨⟩ (f x ∷ map f xs) ʳ++ map f ys ≡⟨⟩ map f (x ∷ xs) ʳ++ map f ys ∎ -- A foldr after a reverse is a foldl. foldr-ʳ++ : ∀ (f : A → B → B) b xs {ys} → foldr f b (xs ʳ++ ys) ≡ foldl (flip f) (foldr f b ys) xs foldr-ʳ++ f b [] {_} = refl foldr-ʳ++ f b (x ∷ xs) {ys} = begin foldr f b ((x ∷ xs) ʳ++ ys) ≡⟨⟩ foldr f b (xs ʳ++ x ∷ ys) ≡⟨ foldr-ʳ++ f b xs ⟩ foldl (flip f) (foldr f b (x ∷ ys)) xs ≡⟨⟩ foldl (flip f) (f x (foldr f b ys)) xs ≡⟨⟩ foldl (flip f) (foldr f b ys) (x ∷ xs) ∎ -- A foldl after a reverse is a foldr. foldl-ʳ++ : ∀ (f : B → A → B) b xs {ys} → foldl f b (xs ʳ++ ys) ≡ foldl f (foldr (flip f) b xs) ys foldl-ʳ++ f b [] {_} = refl foldl-ʳ++ f b (x ∷ xs) {ys} = begin foldl f b ((x ∷ xs) ʳ++ ys) ≡⟨⟩ foldl f b (xs ʳ++ x ∷ ys) ≡⟨ foldl-ʳ++ f b xs ⟩ foldl f (foldr (flip f) b xs) (x ∷ ys) ≡⟨⟩ foldl f (f (foldr (flip f) b xs) x) ys ≡⟨⟩ foldl f (foldr (flip f) b (x ∷ xs)) ys ∎ ------------------------------------------------------------------------ -- reverse -- reverse of cons is snoc of reverse. unfold-reverse : ∀ (x : A) xs → reverse (x ∷ xs) ≡ reverse xs ∷ʳ x unfold-reverse x xs = ʳ++-defn xs -- reverse is an involution with respect to append. reverse-++-commute : (xs ys : List A) → reverse (xs ++ ys) ≡ reverse ys ++ reverse xs reverse-++-commute xs ys = begin reverse (xs ++ ys) ≡⟨⟩ (xs ++ ys) ʳ++ [] ≡⟨ ʳ++-++ xs ⟩ ys ʳ++ xs ʳ++ [] ≡⟨⟩ ys ʳ++ reverse xs ≡⟨ ʳ++-defn ys ⟩ reverse ys ++ reverse xs ∎ -- reverse is involutive. reverse-involutive : Involutive {A = List A} _≡_ reverse reverse-involutive xs = begin reverse (reverse xs) ≡⟨⟩ (xs ʳ++ []) ʳ++ [] ≡⟨ ʳ++-ʳ++ xs ⟩ [] ʳ++ xs ++ [] ≡⟨⟩ xs ++ [] ≡⟨ ++-identityʳ xs ⟩ xs ∎ -- reverse is injective. reverse-injective : ∀ {xs ys : List A} → reverse xs ≡ reverse ys → xs ≡ ys reverse-injective = subst₂ _≡_ (reverse-involutive _) (reverse-involutive _) ∘ cong reverse -- reverse preserves length. length-reverse : ∀ (xs : List A) → length (reverse xs) ≡ length xs length-reverse xs = begin length (reverse xs) ≡⟨⟩ length (xs ʳ++ []) ≡⟨ length-ʳ++ xs ⟩ length xs + 0 ≡⟨ +-identityʳ _ ⟩ length xs ∎ reverse-map-commute : (f : A → B) → map f ∘ reverse ≗ reverse ∘ map f reverse-map-commute f xs = begin map f (reverse xs) ≡⟨⟩ map f (xs ʳ++ []) ≡⟨ map-ʳ++ f xs ⟩ map f xs ʳ++ [] ≡⟨⟩ reverse (map f xs) ∎ reverse-foldr : ∀ (f : A → B → B) b → foldr f b ∘ reverse ≗ foldl (flip f) b reverse-foldr f b xs = foldr-ʳ++ f b xs 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 ------------------------------------------------------------------------ -- _∷ʳ_ module _ {x y : A} where ∷ʳ-injective : ∀ xs ys → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys × x ≡ y ∷ʳ-injective [] [] refl = (refl , refl) ∷ʳ-injective (x ∷ xs) (y ∷ ys) eq with ∷-injective eq ... | refl , eq′ = Prod.map (cong (x ∷_)) id (∷ʳ-injective xs ys eq′) ∷ʳ-injective [] (_ ∷ _ ∷ _) () ∷ʳ-injective (_ ∷ _ ∷ _) [] () ∷ʳ-injectiveˡ : ∀ (xs ys : List A) → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys ∷ʳ-injectiveˡ xs ys eq = proj₁ (∷ʳ-injective xs ys eq) ∷ʳ-injectiveʳ : ∀ (xs ys : List A) → xs ∷ʳ x ≡ ys ∷ʳ y → x ≡ y ∷ʳ-injectiveʳ xs ys eq = proj₂ (∷ʳ-injective xs ys eq) ∷ʳ-++ : ∀ (xs : List A) (a : A) (ys : List A) → xs ∷ʳ a ++ ys ≡ xs ++ a ∷ ys ∷ʳ-++ xs a ys = ++-assoc xs [ a ] ys ------------------------------------------------------------------------ -- DEPRECATED ------------------------------------------------------------------------ -- Please use the new names as continuing support for the old names is -- not guaranteed. -- Version 2.0 zipWith-identityˡ = zipWith-zeroˡ {-# WARNING_ON_USAGE zipWith-identityˡ "Warning: zipWith-identityˡ was deprecated in v2.0. Please use zipWith-zeroˡ instead." #-} zipWith-identityʳ = zipWith-zeroʳ {-# WARNING_ON_USAGE zipWith-identityʳ "Warning: zipWith-identityʳ was deprecated in v2.0. Please use zipWith-zeroʳ instead." #-}