module LiftGet where open import Data.Unit using (⊤ ; tt) open import Data.Nat using (ℕ ; suc) open import Data.Vec using (Vec ; toList ; fromList) renaming ([] to []V ; _∷_ to _∷V_ ; map to mapV) open import Data.List using (List ; [] ; _∷_ ; length ; replicate ; map) open import Data.List.Properties using (length-map) open import Data.Product using (∃ ; _,_ ; proj₁ ; proj₂) open import Function using (_∘_ ; flip ; const) open import Relation.Binary.Core using (_≡_) open import Relation.Binary.PropositionalEquality using (_≗_ ; sym ; cong ; refl ; subst ; trans ; proof-irrelevance ; module ≡-Reasoning) open import Relation.Binary.HeterogeneousEquality using (module ≅-Reasoning ; _≅_ ; ≅-to-≡ ; ≡-to-≅ ; ≡-subst-removable) renaming (refl to het-refl ; sym to het-sym ; cong to het-cong ; reflexive to het-reflexive) import FreeTheorems open import Generic using (length-replicate ; toList-fromList ; toList-subst) open FreeTheorems.ListList using (get-type) renaming (free-theorem to free-theoremL ; Get to GetL ; module Get to GetL) open FreeTheorems.VecVec using () renaming (get-type to getV-type ; Get to GetV ; module Get to GetV) ind-cong : {I : Set} → (X Y : I → Set) → (f : {i : I} → X i → Y i) → {i j : I} → (i ≡ j) → {x : X i} → {y : X j} → x ≅ y → f x ≅ f y ind-cong X Y f refl het-refl = het-refl getVec-to-getList : {getlen : ℕ → ℕ} → (getV-type getlen) → get-type getVec-to-getList get = toList ∘ get ∘ fromList fromList∘map : {α β : Set} → (f : α → β) → (l : List α) → fromList (map f l) ≅ mapV f (fromList l) fromList∘map f [] = het-refl fromList∘map f (x ∷ xs) = ind-cong (Vec _) (Vec _ ∘ suc) (_∷V_ (f x)) (length-map f xs) (fromList∘map f xs) toList∘map : {α β : Set} {n : ℕ} → (f : α → β) → (v : Vec α n) → toList (mapV f v) ≡ map f (toList v) toList∘map f []V = refl toList∘map f (x ∷V xs) = cong (_∷_ (f x)) (toList∘map f xs) GetV-to-GetL : GetV → GetL GetV-to-GetL getrecord = record { get = toList ∘ get ∘ fromList; free-theorem = ft } where open GetV getrecord open ≡-Reasoning ft : {α β : Set} → (f : α → β) → (xs : List α) → toList (get (fromList (map f xs))) ≡ map f (toList (get (fromList xs))) ft f xs = begin toList (get (fromList (map f xs))) ≡⟨ ≅-to-≡ (ind-cong (Vec _) (const (List _)) (toList ∘ get) (length-map f xs) (fromList∘map f xs)) ⟩ toList (get (mapV f (fromList xs))) ≡⟨ cong toList (free-theorem f (fromList xs)) ⟩ toList (mapV f (get (fromList xs))) ≡⟨ toList∘map f (get (fromList xs)) ⟩ map f (toList (get (fromList xs))) ∎ getList-to-getlen : get-type → ℕ → ℕ getList-to-getlen get = length ∘ get ∘ flip replicate tt replicate-length : {A : Set} → (l : List A) → map (const tt) l ≡ replicate (length l) tt replicate-length [] = refl replicate-length (_ ∷ l) = cong (_∷_ tt) (replicate-length l) getList-length : (get : get-type) → {B : Set} → (l : List B) → length (get l) ≡ getList-to-getlen get (length l) getList-length get l = begin length (get l) ≡⟨ sym (length-map (const tt) (get l)) ⟩ length (map (const tt) (get l)) ≡⟨ cong length (sym (free-theoremL get (const tt) l)) ⟩ length (get (map (const tt) l)) ≡⟨ cong (length ∘ get) (replicate-length l) ⟩ length (get (replicate (length l) tt)) ∎ where open ≡-Reasoning length-toList : {A : Set} {n : ℕ} → (v : Vec A n) → length (toList v) ≡ n length-toList []V = refl length-toList (x ∷V xs) = cong suc (length-toList xs) getList-to-getVec-length-property : (get : get-type) → {C : Set} → {m : ℕ} → (v : Vec C m) → length (get (toList v)) ≡ length (get (replicate m tt)) getList-to-getVec-length-property get {_} {m} v = begin length (get (toList v)) ≡⟨ getList-length get (toList v) ⟩ length (get (replicate (length (toList v)) tt)) ≡⟨ cong (length ∘ get ∘ flip replicate tt) (length-toList v) ⟩ length (get (replicate m tt)) ∎ where open ≡-Reasoning getList-to-getVec : get-type → ∃ λ (getlen : ℕ → ℕ) → (getV-type getlen) getList-to-getVec get = getlen , get' where getlen : ℕ → ℕ getlen = getList-to-getlen get get' : {C : Set} {m : ℕ} → Vec C m → Vec C (getlen m) get' {C} v = subst (Vec C) (getList-to-getVec-length-property get v) (fromList (get (toList v))) private module GetV-Implementation (getrecord : GetL) where open GetL getrecord getlen = length ∘ get ∘ flip replicate tt length-property : {C : Set} {m : ℕ} → (s : Vec C m) → length (get (toList s)) ≡ getlen m length-property = getList-to-getVec-length-property get getV : {C : Set} {m : ℕ} → Vec C m → Vec C (getlen m) getV s = subst (Vec _) (length-property s) (fromList (get (toList s))) ft : {α β : Set} (f : α → β) {n : ℕ} (v : Vec α n) → getV (mapV f v) ≡ mapV f (getV v) ft f v = ≅-to-≡ (begin subst (Vec _) (length-property (mapV f v)) (fromList (get (toList (mapV f v)))) ≅⟨ ≡-subst-removable (Vec _) (length-property (mapV f v)) (fromList (get (toList (mapV f v)))) ⟩ fromList (get (toList (mapV f v))) ≅⟨ het-cong (fromList ∘ get) (het-reflexive (toList∘map f v)) ⟩ fromList (get (map f (toList v))) ≅⟨ het-cong fromList (het-reflexive (free-theorem f (toList v))) ⟩ fromList (map f (get (toList v))) ≅⟨ fromList∘map f (get (toList v)) ⟩ mapV f (fromList (get (toList v))) ≅⟨ ind-cong (Vec _) (Vec _) (mapV f) (length-property v) (het-sym (≡-subst-removable (Vec _) (length-property v) (fromList (get (toList v))))) ⟩ mapV f (subst (Vec _) (length-property v) (fromList (get (toList v)))) ∎) where open ≅-Reasoning GetL-to-GetV : GetL → GetV GetL-to-GetV getrecord = record { getlen = getlen; get = getV; free-theorem = ft } where open GetV-Implementation getrecord get-commut-1-≅ : (get : get-type) {A : Set} → (l : List A) → fromList (get l) ≅ proj₂ (getList-to-getVec get) (fromList l) get-commut-1-≅ get l = begin fromList (get l) ≅⟨ het-cong (fromList ∘ get) (≡-to-≅ (sym (toList-fromList l))) ⟩ fromList (get (toList (fromList l))) ≅⟨ het-sym (≡-subst-removable (Vec _) (getList-to-getVec-length-property get (fromList l)) (fromList (get (toList (fromList l))))) ⟩ subst (Vec _) (getList-to-getVec-length-property get (fromList l)) (fromList (get (toList (fromList l)))) ∎ where open ≅-Reasoning get-commut-1 : (get : get-type) {A : Set} → (l : List A) → fromList (get l) ≡ subst (Vec A) (sym (getList-length get l)) (proj₂ (getList-to-getVec get) (fromList l)) get-commut-1 get {A} l = ≅-to-≡ (begin fromList (get l) ≅⟨ get-commut-1-≅ get l ⟩ proj₂ (getList-to-getVec get) (fromList l) ≅⟨ het-sym (≡-subst-removable (Vec _) (sym (getList-length get l)) (proj₂ (getList-to-getVec get) (fromList l))) ⟩ subst (Vec _) (sym (getList-length get l)) (proj₂ (getList-to-getVec get) (fromList l)) ∎) where open ≅-Reasoning get-trafo-1 : (get : get-type) → {B : Set} → getVec-to-getList (proj₂ (getList-to-getVec get)) {B} ≗ get {B} get-trafo-1 get {B} l = begin getVec-to-getList (proj₂ (getList-to-getVec get)) l ≡⟨ refl ⟩ toList (proj₂ (getList-to-getVec get) (fromList l)) ≡⟨ refl ⟩ toList (subst (Vec B) (getList-to-getVec-length-property get (fromList l)) (fromList (get (toList (fromList l))))) ≡⟨ toList-subst (fromList (get (toList (fromList l)))) (getList-to-getVec-length-property get (fromList l)) ⟩ toList (fromList (get (toList (fromList l)))) ≡⟨ toList-fromList (get (toList (fromList l))) ⟩ get (toList (fromList l)) ≡⟨ cong get (toList-fromList l) ⟩ get l ∎ where open ≡-Reasoning GetLVL-identity : (G : GetL) → {A : Set} → GetL.get (GetV-to-GetL (GetL-to-GetV G)) ≗ GetL.get G {A} GetLVL-identity G = get-trafo-1 (GetL.get G) vec-len : {A : Set} {n : ℕ} → Vec A n → ℕ vec-len {_} {n} _ = n fromList-toList : {A : Set} {n : ℕ} → (v : Vec A n) → fromList (toList v) ≅ v fromList-toList []V = het-refl fromList-toList (x ∷V xs) = ind-cong (Vec _) (Vec _ ∘ suc) (_∷V_ x) (length-toList xs) (fromList-toList xs) get-commut-2 : {getlen : ℕ → ℕ} → (get : getV-type getlen) → {B : Set} {n : ℕ} → (toList ∘ get {B} {n}) ≗ (getVec-to-getList get) ∘ toList get-commut-2 get {B} v = ≅-to-≡ (ind-cong (Vec _) (const (List _)) (toList ∘ get) (sym (length-toList v)) (het-sym (fromList-toList v))) get-trafo-2-getlen : {getlen : ℕ → ℕ} → (get : getV-type getlen) → proj₁ (getList-to-getVec (getVec-to-getList get)) ≗ getlen get-trafo-2-getlen {getlen} get n = begin proj₁ (getList-to-getVec (getVec-to-getList get)) n ≡⟨ refl ⟩ length (toList (get (fromList (replicate n tt)))) ≡⟨ length-toList (get (fromList (replicate n tt))) ⟩ vec-len (get (fromList (replicate n tt))) ≡⟨ cong getlen (length-replicate n) ⟩ getlen n ∎ where open ≡-Reasoning get-trafo-2-get-≅ : {getlen : ℕ → ℕ} → (get : getV-type getlen) → {B : Set} {n : ℕ} → (v : Vec B n) → proj₂ (getList-to-getVec (getVec-to-getList get)) v ≅ get v get-trafo-2-get-≅ {getlen} get v = begin subst (Vec _) (getList-to-getVec-length-property (getVec-to-getList get) v) (fromList (toList (get (fromList (toList v))))) ≅⟨ ≡-subst-removable (Vec _) (getList-to-getVec-length-property (getVec-to-getList get) v) (fromList (toList (get (fromList (toList v))))) ⟩ fromList (toList (get (fromList (toList v)))) ≅⟨ fromList-toList (get (fromList (toList v))) ⟩ get (fromList (toList v)) ≅⟨ ind-cong (Vec _) (Vec _ ∘ getlen) get (length-toList v) (fromList-toList v) ⟩ get v ∎ where open ≅-Reasoning get-trafo-2-get : {getlen : ℕ → ℕ} → (get : getV-type getlen) → {B : Set} {n : ℕ} → proj₂ (getList-to-getVec (getVec-to-getList get)) ≗ subst (Vec B) (sym (get-trafo-2-getlen get n)) ∘ get get-trafo-2-get get v = ≅-to-≡ (begin proj₂ (getList-to-getVec (getVec-to-getList get)) v ≅⟨ get-trafo-2-get-≅ get v ⟩ get v ≅⟨ het-sym (≡-subst-removable (Vec _) (sym (get-trafo-2-getlen get (vec-len v))) (get v)) ⟩ subst (Vec _) (sym (get-trafo-2-getlen get (vec-len v))) (get v) ∎) where open ≅-Reasoning