blob: b2273f092948d0bf355aa7098eb6c669c64535e4 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
|
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_)
open import Data.List using (List ; [] ; _∷_ ; length ; replicate ; map)
open import Data.List.Properties using (length-map)
open import Data.Product using (∃ ; _,_ ; proj₂)
open import Function using (_∘_ ; flip ; const)
open import Relation.Binary.Core using (_≡_)
open import Relation.Binary.PropositionalEquality using (_≗_ ; sym ; cong ; refl)
open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎)
get-type : Set₁
get-type = {A : Set} → List A → List A
getV-type : (ℕ → ℕ) → Set₁
getV-type getlen = {A : Set} {n : ℕ} → Vec A n → Vec A (getlen n)
getVec-to-getList : {getlen : ℕ → ℕ} → (getV-type getlen) → get-type
getVec-to-getList get = toList ∘ get ∘ fromList
getList-to-getlen : get-type → ℕ → ℕ
getList-to-getlen get = length ∘ get ∘ flip replicate tt
postulate
free-theorem-list-list : {β γ : Set} → (get : get-type) → (f : β → γ) → get ∘ map f ≗ map f ∘ get
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-theorem-list-list get (const tt) l)) ⟩
length (get (map (const tt) l))
≡⟨ cong (length ∘ get) (replicate-length l) ⟩
length (get (replicate (length l) tt)) ∎
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)
vec-length : {A : Set} {n m : ℕ} → n ≡ m → Vec A n → Vec A m
vec-length refl v = v
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
length-prop : {C : Set} → (m : ℕ) → (v : Vec C m) → length (get (toList v)) ≡ length (get (replicate m tt))
length-prop 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)) ∎
get' : {C : Set} {m : ℕ} → Vec C m → Vec C (getlen m)
get' {_} {m} v = vec-length (length-prop m v) (fromList (get (toList v)))
get-trafo-1 : (get : get-type) → {B : Set} → getVec-to-getList (proj₂ (getList-to-getVec get)) {B} ≗ get {B}
get-trafo-1 get l = begin
getVec-to-getList (proj₂ (getList-to-getVec get)) l
≡⟨ {!!} ⟩
get l ∎
|
