blob: 83a2fe3da9168d56af202477aeee5e92f98e36cd (
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
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
|
{-# OPTIONS --without-K --safe #-}
module Data.Opaque.List where
import Data.List as L
import Function.Construct.Constant as Const
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
open import Algebra.Bundles using (Monoid)
open import Algebra.Morphism using (IsMonoidHomomorphism)
open import Data.List.Effectful.Foldable using (foldable; ++-homo)
open import Data.List.Relation.Binary.Pointwise as PW using (++⁺; map⁺)
open import Data.Product using (_,_; curry′; uncurry′)
open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
open import Data.Setoid using (∣_∣)
open import Data.Unit.Polymorphic using (⊤)
open import Effect.Foldable using (RawFoldable)
open import Function using (_⟶ₛ_; Func; _⟨$⟩_; id)
open import Level using (Level; _⊔_)
open import Relation.Binary using (Setoid)
open Func
private
variable
a c ℓ : Level
A B : Set a
Aₛ Bₛ : Setoid c ℓ
⊤ₛ : Setoid c ℓ
⊤ₛ = record { Carrier = ⊤ ; _≈_ = λ _ _ → ⊤ }
opaque
List : Set a → Set a
List = L.List
[] : List A
[] = L.[]
_∷_ : A → List A → List A
_∷_ = L._∷_
map : (A → B) → List A → List B
map = L.map
_++_ : List A → List A → List A
_++_ = L._++_
Listₛ : Setoid c ℓ → Setoid c (c ⊔ ℓ)
Listₛ = PW.setoid
[]ₛ : ⊤ₛ {c} {c ⊔ ℓ} ⟶ₛ Listₛ {c} {ℓ} Aₛ
[]ₛ = Const.function ⊤ₛ (Listₛ _) []
∷ₛ : Aₛ ×ₛ Listₛ Aₛ ⟶ₛ Listₛ Aₛ
∷ₛ .to = uncurry′ _∷_
∷ₛ .cong = uncurry′ PW._∷_
[-]ₛ : Aₛ ⟶ₛ Listₛ Aₛ
[-]ₛ .to = L.[_]
[-]ₛ .cong y = y PW.∷ PW.[]
mapₛ : (Aₛ ⟶ₛ Bₛ) → Listₛ Aₛ ⟶ₛ Listₛ Bₛ
mapₛ f .to = map (to f)
mapₛ f .cong xs≈ys = map⁺ (to f) (to f) (PW.map (cong f) xs≈ys)
cartesianProduct : ∣ Listₛ Aₛ ∣ → ∣ Listₛ Bₛ ∣ → ∣ Listₛ (Aₛ ×ₛ Bₛ) ∣
cartesianProduct = L.cartesianProduct
cartesian-product-cong
: {xs xs′ : ∣ Listₛ Aₛ ∣}
{ys ys′ : ∣ Listₛ Bₛ ∣}
→ (let open Setoid (Listₛ Aₛ) in xs ≈ xs′)
→ (let open Setoid (Listₛ Bₛ) in ys ≈ ys′)
→ (let open Setoid (Listₛ (Aₛ ×ₛ Bₛ)) in cartesianProduct xs ys ≈ cartesianProduct xs′ ys′)
cartesian-product-cong PW.[] ys≋ys′ = PW.[]
cartesian-product-cong {Aₛ = Aₛ} {Bₛ = Bₛ} {xs = x₀ L.∷ xs} {xs′ = x₀′ L.∷ xs′} (x₀≈x₀′ PW.∷ xs≋xs′) ys≋ys′ =
++⁺
(map⁺ (x₀ ,_) (x₀′ ,_) (PW.map (x₀≈x₀′ ,_) ys≋ys′))
(cartesian-product-cong {Aₛ = Aₛ} {Bₛ = Bₛ} xs≋xs′ ys≋ys′)
pairsₛ : Listₛ Aₛ ×ₛ Listₛ Bₛ ⟶ₛ Listₛ (Aₛ ×ₛ Bₛ)
pairsₛ .to = uncurry′ L.cartesianProduct
pairsₛ {Aₛ = Aₛ} {Bₛ = Bₛ} .cong = uncurry′ (cartesian-product-cong {Aₛ = Aₛ} {Bₛ = Bₛ})
++ₛ : Listₛ Aₛ ×ₛ Listₛ Aₛ ⟶ₛ Listₛ Aₛ
++ₛ .to = uncurry′ _++_
++ₛ .cong = uncurry′ ++⁺
foldr : (Aₛ ×ₛ Bₛ ⟶ₛ Bₛ) → ∣ Bₛ ∣ → ∣ Listₛ Aₛ ∣ → ∣ Bₛ ∣
foldr f = L.foldr (curry′ (to f))
foldr-cong
: (f : Aₛ ×ₛ Bₛ ⟶ₛ Bₛ)
→ (e : ∣ Bₛ ∣)
→ (let module [A]ₛ = Setoid (Listₛ Aₛ))
→ {xs ys : ∣ Listₛ Aₛ ∣}
→ (xs [A]ₛ.≈ ys)
→ (open Setoid Bₛ)
→ foldr f e xs ≈ foldr f e ys
foldr-cong {Bₛ = Bₛ} f e PW.[] = Setoid.refl Bₛ
foldr-cong f e (x≈y PW.∷ xs≋ys) = cong f (x≈y , foldr-cong f e xs≋ys)
foldrₛ : (Aₛ ×ₛ Bₛ ⟶ₛ Bₛ) → ∣ Bₛ ∣ → Listₛ Aₛ ⟶ₛ Bₛ
foldrₛ f e .to = foldr f e
foldrₛ {Bₛ = Bₛ} f e .cong = foldr-cong f e
module _ (M : Monoid c ℓ) where
open Monoid M renaming (setoid to Mₛ)
opaque
unfolding List
fold : ∣ Listₛ Mₛ ∣ → ∣ Mₛ ∣
fold = RawFoldable.fold foldable rawMonoid
fold-cong
: {xs ys : ∣ Listₛ Mₛ ∣}
→ (let module [M]ₛ = Setoid (Listₛ Mₛ))
→ (xs [M]ₛ.≈ ys)
→ fold xs ≈ fold ys
fold-cong = PW.rec (λ {xs} {ys} _ → fold xs ≈ fold ys) ∙-cong refl
foldₛ : Listₛ Mₛ ⟶ₛ Mₛ
foldₛ .to = fold
foldₛ .cong = fold-cong
opaque
unfolding fold
++ₛ-homo
: (xs ys : ∣ Listₛ Mₛ ∣)
→ foldₛ ⟨$⟩ (++ₛ ⟨$⟩ (xs , ys)) ≈ (foldₛ ⟨$⟩ xs) ∙ (foldₛ ⟨$⟩ ys)
++ₛ-homo xs ys = ++-homo M id xs
[]ₛ-homo : foldₛ ⟨$⟩ ([]ₛ ⟨$⟩ _) ≈ ε
[]ₛ-homo = refl
module _ (M N : Monoid c ℓ) where
module M = Monoid M
module N = Monoid N
open IsMonoidHomomorphism
opaque
unfolding fold
fold-mapₛ
: (f : M.setoid ⟶ₛ N.setoid)
→ IsMonoidHomomorphism M.rawMonoid N.rawMonoid (to f)
→ {xs : ∣ Listₛ M.setoid ∣}
→ foldₛ N ⟨$⟩ (mapₛ f ⟨$⟩ xs) N.≈ f ⟨$⟩ (foldₛ M ⟨$⟩ xs)
fold-mapₛ f isMH {L.[]} = N.sym (ε-homo isMH)
fold-mapₛ f isMH {x L.∷ xs} = begin
f′ x ∙ fold N (map f′ xs) ≈⟨ N.∙-cong N.refl (fold-mapₛ f isMH {xs}) ⟩
f′ x ∙ f′ (fold M xs) ≈⟨ homo isMH x (fold M xs) ⟨
f′ (x ∘ fold M xs) ∎
where
open N using (_∙_)
open M using () renaming (_∙_ to _∘_)
open ≈-Reasoning N.setoid
f′ : M.Carrier → N.Carrier
f′ = to f
|
